A '(C)' marks a related Colloquium Talk at 4:30 PM.
The asterisks (*) mark meetings of the QUANTUM MATH SEMINAR, which has occasionally replaced the algebra seminar, during Spring 1998-Spring 2008.
The sharp (#) marks a meeting of the VIGRE seminar, which occasionally replaced the algebra seminar during 2000-2002.

Click here for the algebra seminars in the current semester

Spring 2018 Seminars (Wednesdays at 2:00 in H425)
24 Jan Aurélien Sagnier EP-Palaiseau "An arithmetic site of Connes-Consani type for the Gaussian integers"
31 Jan Jozsef Beck   Rutgers   "An annoying question about eigenvalues"
 7 Feb Lev Borisov  Rutgers    "Equations of Cartwright-Steger surface"
21 Feb Dhruv Ranganathan  MIT  "Curves, maps, and singularities in genus one"
28 Feb Rohini Ramadas  Harvard "Algebraic dynamics from topological and holomorphic dynamics"
15 Mar Nicola Pagani U.Liverpool "The indeterminacy of universal Abel-Jacobi sections" 
28 Mar Ana-Maria Castravet Northeastern U. "Derived categories of moduli spaces of stable rational curves"
11 Apr Chiara Damiolini Rutgers "Conformal blocks attached to twisted groups" 
18 Apr Joe Waldron  Princeton  "Singularities of general fibers in positive characteristic"
 2 May Ben Bakker  U. Georgia  "Hodge theory and o-minimal geometry"
 9 May Antonella Grassi U. Penn "Singularities in geometry, topology and strings"  (in H705)
Classes end Monday, April 30; Final Exams are May 4-9, 2018
Fall 2017 Seminars (Wednesdays at 2:00 in H525)
13 Sep Louis Rowen  Bar-Ilan Univ "A general algebraic structure theory for tropical mathematics"
20 Sep Nicola Tarasca Rutgers   "K-classes of Brill-Noether loci and a determinantal formula"
27 Sep Pham Huu Tiep  Rutgers   "Character levels and character bounds"
4 Oct Dave Jensen     Yale      "Linear Systems on General Curves of Fixed Gonality"
11 Oct Gernot Stroth  Martin-Luther Univ.  "On the Thompson Subgroup"
18 Oct Han-Bom Moon   IAS      "Birational geometry of moduli spaces of parabolic bundles"
 1 Nov Danny Krashen  Rutgers   "Extremely indecomposable division algebras"
 8 Nov Lev Borisov    Rutgers   "Explicit equations of a fake projective plane"
15 Nov Julia Hartmann U. Penn.  "Local-global principles for rational points and zero-cycles"
22 Nov --- no seminar ---      Thanksgiving is Nov. 23; Friday class schedule 
29 Nov Chuck Weibel   Rutgers   "K-theory of line bundles and smooth varieties"
 6 Dec Seth Baldwin  N.Carolina "Equivariant K-theory associated to Kac-Moody groups"
13 Dec Brooke Ullery  Harvard   "Gonality of complete intersection curves"
Classes end December 13; Final Exams are December 15-22, 2017

Spring 2017 Seminars (Wednesdays at 2:00 in H705)
22 Feb Ryan Shifler  Virginia Tech "Equivariant Quantum Cohomology of the Odd Symplectic Grassmannian"
 1 Mar Chuck Weibel    Rutgers   "The Witt group of surfaces and 3-folds"
 8 Mar Oliver Pechenik Rutgers   "Decompositions of Grothendieck polynomials"
15 Mar no seminar      ------------------- Spring Break ----------
22 Mar Ilya Kapovich UIUC/Hunter College "Dynamics and polynomial invariants for free-by-cyclic groups"
29 Mar Rachel Levanger Rutgers "Interleaved persistence modules and applications of persistent homology to problems in fluid dynamics"
 5 Apr Cristian Lenart Albany-SUNY "Kirillov-Reshetikhin modules and Macdonald polynomials: a survey and applications"
19 Apr Anders Buch     Rutgers     "Puzzles in quantum Schubert calculus"
26 Apr Sjuvon Chung    Rutgers     "Equivariant quantum K-theory of projective space"
Classes end May 1; Final Exams are May 4-10, 2017
Fall 2016 Seminars (Wednesdays at 2:00 in H423)
21 Sept Fei Qi          Rutgers  "What is a meromorphic open string vertex algebra?"
28 Sept Zhuohui Zhang   Rutgers  "Quaternionic Discrete Series"
 5 Oct  Sjuvon Chung    Rutgers  "Euler characteristics in cominuscule quantum K-theory"
12 Oct  Ed Karasiewicz  Rutgers  "Elliptic Curves and Modular Forms"
19 Oct  Natalie Hobson U.Georgia "Quantum Kostka and the rank one problem for sl2m"
26 Oct Oliver Pechenik Rutgers   "K-theoretic Schubert calculus"
 2 Nov Vasily Dolgushev Temple U "The Intricate Maze of Graph Complexes"
 9 Nov Jason McCullough Rider U. "Rees-like Algebras and the Eisenbud-Goto Conjecture"
16 Nov Robert Laugwitz  Rutgers  "Representations of p-DG 2-categories"
23 Nov --- no seminar ---      Thanksgiving is Nov. 24; Friday class schedule 
30 Nov Semeon Artamonov Rutgers  "Double Gerstenhaber algebras of noncommutative poly-vector fields"
 7 Dec Daniel Krashen  U.Georgia "Geometry and the arithmetic of algebraic structures" (Special talk)
14 Dec Angela Gibney  U.Georgia  "Vector bundles of conformal blocks on the moduli space of curves" (Special talk)
Classes end December 14; Final Exams are December 16-23, 2016

Spring 2016 Seminars (Wednesdays at 2:00 in H705)
20 Jan Louis Rowen  Bar-Ilan Univ "Symmetrization in tropical algebra"
 3 Feb Volodia Retakh   Rutgers   "Generalized adjoint actions"  
10 Feb Omer Bobrowski   Duke (@noon!) "Random Topology and its Applications"
17 Feb Lisa Carbone     Rutgers   "Arithmetic constructions of hyperbolic Kac-Moody groups"
 2 Mar Chuck Weibel     Rutgers   "Relative Cartier divisors"
 9 Mar Lev Borisov      Rutgers   "Elliptic genera of singular varieties and related topics"
16 Mar  no seminar      ------------------- Spring Break ----------
23 Mar Rachel Levanger  Rutgers   "Auslander-Reiten quivers of finite-dimensional algebras"
30 Mar Richard Lyons    Rutgers   "Aspects of the Classification of simple groups"
 6 Apr Richard Lyons    Rutgers   "Aspects of the Classification, continued"
13 Apr Siddhartha Sahi  Rutgers   "Eigenvalues of generalized Capelli operators"
20 Apr Ed Karasiewicz   Rutgers   "Some Aspects of p-adic Representations & the Casselman-Shalika Formula" 
27 Apr Semeon Artamonov Rutgers   "Noncommutative Poisson Geometry"
Classes end May 2; Final Exams are May 4-10

Fall 2015 Seminars (Wednesdays at 2:00 in H425)
 7 Oct Chuck Weibel Rutgers   "Monoids, monoid rings and monoid schemes"
14 Oct Lev Borisov  Rutgers   "Introduction to A-D-E singularities"
21 Oct Dylan Allegretti  Yale "Quantization of Fock and Goncharov's canonical basis"
28 Oct Volodia Retakh Rutgers "Noncommutative Cross Ratios"
 4 Nov Gabriele Nebe  U.Aachen "Automorphisms of extremal codes"
11 Nov Chuck Weibel Rutgers   "Relative Cartier divisors and polynomials"
18 Nov Glen Wilson  Rutgers   "Motivic stable homotopy over finite fields"
25 Nov --- no seminar ---      Thanksgiving is Nov. 26; Friday class schedule 
 2 Dec Anders Buch  Rutgers   "The Thom Porteous formula"
 9 Dec Pham Huu Tiep  U. Arizona "Representations of finite groups and applications "
Classes end Dec. 10; Final Exams are December 15-22

Spring 2015 Seminars (Wednesdays at 2:00 in H124)
27 Jan ---
 4 Feb Jesse Wolfson Chicago  "The Index Map and Reciprocity Laws for Contou-Carrère Symbols"
18 Feb Justin Lynd   Rutgers  "Fusion systems and centric linking systems"
25 Feb Lev Borisov   Rutgers  "Zero divisors in the Grothendieck ring of varieties"
 4 Mar Volodia Retakh Rutgers "Noncommutative triangulations and the Laurent phenomenon"
 6 MarC Burt Totaro  UCLA/IAS "Birational geometry and algebraic cycles" (Colloquium)
11 Mar Anders Buch   Rutgers  "TK"
18 Mar no seminar      ------------------- Spring Break ------------------ 
22 Apr Howard Neuer  Rutgers  "On special cubic 4-folds"
Classes end May 4; Spring Final Exams are May 7-13

Fall 2014 Seminars (Wednesdays, 3:15-4:15PM in H525)
17 Sep Edwin Beggs  U.Swansea "Semiclassical approximation to noncommutative Riemannian geometry"
24 Sep Anders Buch   Rutgers  "Equivariant quantum cohomology and puzzles"
 8 Oct Lev Borisov   Rutgers  "Cut and paste approaches to rationality of cubic fourfolds"
15 Oct Chuck Weibel  Rutgers  "The Witt group of real varieties"
22 Oct Ed Karasiewicz Rutgers "Jacobians of modular curves"
29 Oct Charlie Siegel (IPMU Japan) "A Modular Operad of Embedded Curves"
 5 Nov no seminar
12 Nov Marvin Tretkoff Texas A&M "Some non-compact Riemann surfaces branched over three points"
19 Nov Ashley Rall  U. Virginia "Property T for Kac-Moody groups"
26 Nov (Thanksgiving is Nov. 27)   no seminar
 3 Dec Alex Lubotzky NYU/Hebrew U. (Israel) "Sieve methods in group theory"

Spring 2014 Seminars (Wednesdays at 2:00 in H124)
Apr 26 Anatoly Vershik, St. Petersburg State University, Russia "Invariant measures and standardness"
Mar  5 Greg Muller, Michigan "Locally acyclic cluster algebras"
Mar 12 Julianne Rainbolt, Saint Louis University  "Bruhat cells which contain only regular elements"
Mar 26 Bianca Viray, Brown U "Unramified Brauer classes on cyclic covers of the projective plane"
Apr 9 Lev Borisov, Rutgers "An annoying problem in toric geometry"
Apr 23 Howard Nuer, Rutgers "An introduction to cubic fourfolds and their moduli space"
Apr 30 Vijay Ravikumar, Tata Institute  "Equivariant Pieri rules for Isotropic Grassmannians"

Fall 2013 Seminars (Wednesdays at 2:00 in H525)
Details for Fall 2013 seminars are located at THIS SITE
 4 Sep Delaram Kahrobaei CUNY   "Applications of Algebra in Information Security"
 2 Oct Bob Guralnick USC and IAS  "Dimensions of Fixed Spaces"
 9 Oct Leonid Petrov Northeastern "Robinson-Schensted-Knuth correspondences and their $(q,t)$-deformations"
16 Oct Knight Fu       Rutgers  "Torsion Theory and Slice Filtration of Homotopy Invariant Sheaves With Transfers"
23 Oct Ralph Kaufmann  Purdue/IAS "Three Hopf algebras and their common algebraic and categorical background"
30 Oct Howard Nuer     Rutgers  "Bridgeland Stability and Moduli on Enriques Surfaces"
 6 Nov Andrew Blumberg  U.Texas  "Probabilistic inference in topological data analysis"
13 Nov Pierre Cartier  IHES     "Galois groups of differential equations: a noncommutative analog"
20 Nov Zsolt Patakfalvi Princeton "Classification of algebraic varieties: classical results and recent advances in positive characteristic"

Spring 2013 Seminars (Wednesdays at 2:00 in H525)
24 Jan Daniel Erman    Michigan "Equations, syzygies, and vector bundles"
30 Jan David Anderson  U. Paris "Equivariant Schubert calculus: positivity, formulas, applications"
 6 Feb Chuck Weibel    Rutgers  "What is a Derivator?"
13 Feb V. Retakh       Rutgers  "A geometric approach to noncommutative Laurent phenomenon"
20 Feb Tatiana Bandman Bar-Ilan "Dynamics and surjectivity of some
word maps on SL(2,q)" 27 Feb Bob Guralnick USC and IAS "Strongly Dense Subgroups of Algebraic Groups" 13 Mar Mina Teicher Bar-Ilan "The 3 main problems in the braid group" 20 Mar no seminar -------------- Spring Break ------------- 3 Apr Joe Ross USC "Intersection theory on singular varieties" 10 Apr Lev Borisov Rutgers "Hilbert modular threefolds of discriminant 49" 17 Apr Charlie Siegel (IPMU Japan) "Cyclic Covers, Prym Varieties and the Schottky-Jung Relations" 24 Apr Freya Pritchard CUNY "Implicit systems of differential equations" 1 May Alexei Stepanov (St.Petersburg State University) "Structure of Chevalley groups over rings"

Fall 2012 Seminars (Wednesdays at 2:00 in H525)
19 Sept Chuck Weibel Rutgers "Binary codes and Galois covers of varieties"
10 Oct Anders Buch  Rutgers "Curve neighborhoods"
17 Oct Dan Grayson  IAS  "Computations in intersection theory"
24 Oct Justin Lynd  Rutgers "Fusion systems with prescribed involution centralizers"
31 Oct Lev Borisov  Rutgers  "On Hilbert modular threefolds of discriminant 49"
 7 Nov Oliver Rondigs Osnabruck, Germany "On the slice filtration for hermitian K-theory"
14 Nov Howie Nuer   Rutgers  "surfaces on Calabi-Yao 3-folds"
21 Nov  no seminar, Friday classes	(Thanksgiving week)
28 Nov Susan Durst  Rutgers  "Universal labelling algebras for directed graphs"
 5 Dec Anastasia Stavrova U.Essen "Injectivity property of etale H^1, non-stable K_1, and other functors"
12 Dec Joe Ross     USC    "Presheaves with oriented weak transfers"
Fall 2012 Semester starts Sept.4; (Wednesday Nov. 21 will be Friday classes).
Classes end Wed., Dec. 12; Final Exams start Friday 12/14/11

Spring 2012 Seminars (Wednesdays at 2:00 in H525)
25 Jan Vasily Dolgashev  Temple Univ. "Exhausting quantization procedures"
 8 Feb Chuck Weibel Rutgers       "Shift Equivalence and Z[t]-modules"
15 Feb Pablo Pelaez Rutgers	  "An introduction to weights"
22 Feb Anders Buch  Rutgers	  "K-theory of miniscule varieties"
29 Feb Julia Plavnik  U.Cordoba   "From algebra to category theory: a first approach to fusion categories"
 7 Mar Anastasia Stavrova U.Essen "On the unstable K_1-functors associated to simple algebraic groups"
14 Mar no seminar	-------------- Spring Break ------------- 
21 Mar Mark Walker  U. Nebraska   "Invariants of Matrix Factorizations"
28 Mar Lev Borisov  Rutgers       "Combinatorial aspects of toric mirror symmetry"
 5 Apr Joe Ross     USC           "Cohomology Theories with Supports" Thursday 11:00AM in Hill 425
11 Apr V. Retakh    Rutgers       "Noncommutative Laurent Phenomena"
18 Apr Ben Wyser    U.Georgia    "Symmetric subgroup orbit closures on flag varieties as universal degeneracy loci"
25 Apr Ling Bao   Chalmers U. (Sweden)  "Algebraic symmetries in supergravity"
Spring 2012 Semester starts Jan. 17, Classes end April 30,
Spring Break is March 11-18, Exams start May 3.
Fall 2011 Seminars (Wednesdays at 2:00PM in H423)
14 Sept Charles Siegel  U. Penn.  "The Schottky Problem and genus 5 curves"
28 Sept Abid Ali        Rutgers   "Congruence subgrous of lattices in rank 2 Kac-Moody groups over finite fields" 
 5 Oct Raika Dehy  Cergy-Pontoise "Cluster algebras and categorification" 
12 Oct Chuck Weibel    Rutgers    "What (besides varieties) are motivic spaces?"
19 Oct Raika Dehy  Cergy-Pontoise "Cluster algebras and categorification (bis)"
26 Oct Anders Buch     Rutgers    "Giambelli formulas for orthogonal Grassmannians"
 2 Nov Alice Rizzardo  Columbia   "On Fourier-Mukai type functors"
 9 Nov Changlong Zhong  Ottowa    "Comparison of Dualizing Complexes"
16 Nov Anastasia Stavrova U.Essen "The Serre-Grothendieck conjecture on torsors and the
                                    classification of simple algebraic groups"
23 Nov no seminar, no classes		    (Thanksgiving week)
30 Nov Lev Borisov     Rutgers    "Elliptic functions and equations of modular curves"
 7 Dec Pablo Pelaez    Rutgers    "Homotopical Methods in Algebraic Geometry"
Fall 2011 Semester starts Sept.1; (Thursday Sept.8 will be Monday classes).
Classes end Tues, Dec. 13; Final Exams start Friday 12/16/11

Spring 2011 Seminars
(Wednesdays at 2 PM in CoRE 431)
21 Jan Chenyang Xu    Princeton    Colloquium talk (Friday)
26 Jan Grigor Sargsyan UCLA        TBA (Monday Jan. 24)
28 Jan Ivan Losev     MIT          Colloquium talk (Friday)
 4 Feb A. Salehi Golsefidy Princeton Colloquium talk (Friday)
 9 Feb Louis Rowen    Bar Ilan U.  "Tropical Algebra"
16 Feb no seminar
23 Feb Christian Haesemeyer UCLA "Rational points, zero cycles of degree one, and A^1-homotopy theory"
 2 Mar Volodia Retakh  Rutgers    "Linear recursive sequences, Laurent phenomenon and Dynkin diagrams"
 9 Mar Chuck Weibel    Rutgers    "Monoid algebras and monoid schemes"
16 Mar no seminar	-------------- Spring Break ------------- 
30 Mar Volodia Retakh  Rutgers    "Hilbert series of algebras associated to direct graphs and order homology"
 6 Apr Lev Borisov     Rutgers    "Syzygies of binomial ideals and toric Eisenbud-Goto conjecture"
13 Apr Crichton Ogle   Ohio State "Cyclic homology, simplicial rapid decay algebras, and applications to K*t(l¹(G))"
20 Apr Susan Durst     Rutgers    "Twisted Polynomial Rings and Embeddings of the Free Algebra"
27 Apr Chuck Weibel    Rutgers    "Derived categories of graded modules"
 4 May
Spring Finals are May 5-11; last day of classes is May 2 (Monday)

Fall 2010 Seminars (Mondays at 4:30PM in H705)
20 Sept Uma Iyer  Bronx Community College  "Quantum differential operators" (4:50 PM)
27 Sept Chuck Weibel   Rutgers    "Monoids and algebraic geometry" (4:50 PM)
 4 Oct Bob Guralnick  USC         "Dimensions of fixed point spaces of elements in linear groups" (4:50 PM)
11 Oct Volodia Retakh  Rutgers    "Hilbert series of algebras associated to directed graphs and order homology" (4:50 PM)
18 Oct Lev Borisov     Rutgers    "The Pfaffian-Grassmannian derived equivalence" (4:30 PM)
 1 Nov Chuck Weibel    Rutgers    "etale cohomology operations" (4:30 PM)
 8 Nov Anders Buch     Rutgers    "Pieri rules for the K-theory of cominuscule Grassmannians" (4:30 PM)
15 Nov Volodia Retakh  Rutgers    "A short proof of the Kontsevich cluster conjecture" (4:30 PM)
22 Nov  no seminar  (Wednesday class schedule, Thanksgiving week)
29 Nov Earl Taft       Rutgers    "The Lie product in the continuous Lie dual of the Witt algebra" (4:30 PM)
 6 Dec Chuck Weibel    Rutgers    "Motivic cohomology operations" (4:30 PM)
13 Dec Ralph Kaufmann  Purdue&IAS "Algebraic Structures from Operads" (4:30 PM)
Fall Finals are Dec. 16-23; last day of classes is Dec 13 (Monday)

Spring 2010 Seminars (Mondays at 4:50 in H705)
 1 Feb Max Karoubi  Univ. Paris 7 "Periodicity in Hermitian K-groups"
15 Feb Chuck Weibel Rutgers      Exceptional objects (after Polishchuk)
22 Feb
 1 Mar Ray Hoobler  CCNY         "Applications of stable bundles to Witt groups and Brauer groups"
 8 Mar Christian Kassel CNRS & U.Strasbourg "Drinfeld twists and finite groups"
15 Mar no seminar	-------------- Spring Break ------------- 
22 Mar Earl Taft       Rutgers   "Hopf algebras and recursive sequences"
29 Mar Chuck Weibel    Rutgers   "Tilting 1"
 5 Apr Carlo Mazza     U. Genoa  "The K-theory of motives"
12 Apr Miodrag Iovanov  USC      "Generalized Frobenius algebras, Integrals and applications to Hopf algebras and compact groups"
19 Apr Chuck Weibel    Rutgers   "Tilting 2"
26 Apr Robert Wilson   Rutgers   "Tilting 3 "
 3 May William Keigher Rutgers-Newark "Module Structures on the Ring of Hurwitz Series"
Spring Break is March 13-21, 2010; Final Exams begin Thursday May 6. 

Fall 2009 Seminars (joint with Gelfand Seminar)
28 Sep no seminar     Yom Kippur
 5 Oct Lourdes Juan   Texas Tech   Differential Central Simple Algebras and Picard-Vessiot representations
12 Oct Bob Guralnick  USC          Derangements in Finite and Algebraic Groups
19 Oct Ken Johnson    Penn State-Abington Mathematics arising from a new look at the
                                   Dedekind-Frobenius group matrix and group determinant
 2 Nov Chloe Perin    Hebrew Univ. "Induced definable structure on cyclic subgroups of the free group"
 9 Nov Paul Ellis     U. Connecticut "The classfication problem for finite rank dimension groups"
16 Nov Ravi Srinivasan  RU-Newark  "Picard-Vessiot Theory"
23 Nov Vladimir Retakh  Rutgers    "Noncommutative algebra and combinatorial topology"
30 Nov Chuck Weibel     Rutgers    "homotopy model structures as tools for homogical algebra"
 7 Dec no seminar       cancelled due to Gelfand Memorial  

Fall 2009 Semester begins Tuesday Sept 1; Labor Day is Sept. 7
Final Exams begin Wednesday Dec 16, 2009; Math Group Exams are Dec. 16 (4-7PM).
Spring 2009 Algebra/Gelfand Seminar
 2 Feb:  Chuck Weibel	  Rutgers    "Stability conditions for triangulated categories"
 9 Feb: Luis Caffarelli    U. Texas   Special Colloquium talk at this time
16 Feb: Vladimir Retakh    Rutgers    "Lie algebras over noncommutative rings"
23 Feb: Leon Pritchard   CUNY         "Partitioned differential quasifields"
 2 Mar: Jan Manschot  Rutgers-Physics  "Stability conditions in physics"
16 Mar  no seminar	-------------- Spring Break ------------- 
30 Mar: Elizabeth Gasparim Edinburgh  "The Nekrasov Conjecture for Toric Surfaces"
 6 Apr: Vladimir Retakh    Rutgers    "Noncommutative Laurent phenomenon"
13 Apr: Bill Keigher   Rutgers-Newark "Differential quasifields"
20 Apr: Chris Woodward     Rutgers    "Morphisms of cohomological field theories and behavior of Gromov-Witten invariants under quotients"
27 Apr: Gregory Ginot    Univ.Paris   "higher order Hochschild (co)homology"
Spring Break is March 14-22, 2009; Final Exams begin Thursday May 7.

Fall 2008 Algebra/Gelfand Seminar (at 4:00 Mondays)

 5 Sep:# Paul Baum         Penn State "Morita Equivalence Revisited" Talk is Friday at 2PM in H705
15 Sep:   no seminar        MSMF Reception
18 Sep: Vasily Dolgushev UC Riverside "Formality theorems for Hochschild (co)chains and their applications" Talk at 2PM in H425
22 Sep: Mike Zieve           Rutgers  "Rationality and integrality in dynamical systems"
29 Sep  no seminar	    Rosh Hoshanna
 6 Oct: Chuck Weibel	    Rutgers  "The de Rham-Witt complex of R[t]"
13 Oct: Anders Buch          Rutgers  "Quantum K-theory" 
20 Oct: Earl Taft            Rutgers  "Combinatorial Identities and Hopf Algebras"
27 Oct: Siddhartha Sahi      Rutgers  "Interpolation and binomial identities in several variables"
 3 Nov: Leigh Cobbs          Rutgers  "Infinite towers of co-compact lattices in Kac-Moody groups"
10 Nov: Jarden         Logic Seminar  "The absolute Galois group of subfields of the field of totally S-adic numbers"
14 Nov: Guillermo Cortiñas Buenos Aires "K-theory of some algebras associated to quivers" Talk is Friday at 2PM in H425
17 Nov  no seminar          -------   ------------------------------
24 Nov: Robert Wilson        Rutgers   "Splitting Algebras associated to cell complexes"
 1 Dec: Roozbeh Hazrat  Queens Univ. Belfast "Reduced K-theory of Azumaya algebras"
 9 Dec: Steven Duplij   Kharkov Univ. "Quantum Enveloping Algebras and the Pierce Decomposition " Talk is Tuesday, 2PM in H425

Fall 2008 Semester begins Tuesday Sept 2; Final Exams begin Monday Dec 15, 2008.

Spring 2008 Algebra/Gelfand Seminar

The Algebra Seminar was merged with the Gelfand Seminar, and met Mondays at 4:40 PM.
25 Jan(F) W. Vasconcelos Rutgers  The Chern numbers of a local ring (I)
28 Jan: Vladimir Retakh  Rutgers  "Obstructions to formality and obstructions to deformations" 
 4 Feb: Chuck Weibel     Rutgers  "Generation of Galois cohomology by symbols"
 5 Feb(T)* Tony Milas  SUNY Albany "W-algebras, quantum groups and combinatorial identities"
 8 Feb(F) M. Zieve       Rutgers  "The lattice of subfields of K(x)
11 Feb: Zin Arai       Kyoto Univ "Complex dynamics and shift automorphism groups"
18 Feb: Andrzej Zuk    Univ Paris  "Automata Groups"
25 Feb: Mike Zieve       Rutgers  "Automorphism groups of curves"
29 Feb(F) Laura Ghezzi	 CUNY      "Generalizations of the Strong Castelnuovo Lemma"
 3 Mar: Chuck Weibel    Rutgers   "Model categories versus derived categories"
10 Mar:  R Parimala      Emory Univ.  "Rational points on homogeneous spaces"
14 Mar#* Tom Robinson	 Rutgers  "Formal differential representations" 11:55 AM Friday in Hill 425
17 Mar:  no seminar	-------------- Spring Break ------------- 
28 Mar#* David Ben-Zvi  IAS & U.Texas  "Real Groups and Topological Field Theory"
28 Mar(F) Jooyoun Hong   Purdue   "Homology and Elimination"
31 Mar: Siddhartha Sahi Rutgers   "Tensor categories and equivariant cohomology"
 4 Apr(C) David Saltman CCR and U.Texas "Division Algebras over Surfaces"
 7 Apr: Earl Taft       Rutgers   "The boson-fermion correspondence and one-sided quantum groups
14 Apr: Colleen Duffy   Rutgers   "Graded traces and irreducible representations of graph algebras" 
21 Apr: Semyon Alesker Tel-Aviv U. "Plurisubharmonic functions on the octonionic plane and Spin(9)-invariant valuations on convex sets"
28 Apr: Jim Borger  Australia Natl Univ. "Witt vectors, Lambda-rings, and absolute algebraic geometry"
 5 May: Richard Lyons   Rutgers    "Subgroups of Algebraic Groups and Finite Groups"

Spring 2008 Semester begins Tuesday Jan 22; Spring Finals are May 8-14, 2008

Fall 2007 Algebra Seminar

During Fall 2005-Fall 2007, the seminar was rescheduled to 1:00-2:00PM Fridays in H705, as a result of new class times.
 7 Sep* Benjamin Doyon  Durham      Conformal field theory and Schramm-Loewner evolution
14 Sep* Liang Kong      Max Planck  An introduction to open-closed conformal field theory
28 Sep Richard Lyons    Rutgers    Presidential Address and Department Reception
 5 Oct Diane Maclagan  Rutgers-Warwick Starts at 2:15! "Equations for Chow and Hilbert quotients"
12 Oct Rafael Villareal IPN,Mexico  "Unmixed clutters with a perfect matching"
19 Oct  POSTPONED to November 16
2 Nov# Andrea Miller    Harvard   POSTPONED
 9 Nov Dan Krashen       U. Penn   Starts at 2:20!Patching subfields of division algebras
16 Nov Angela Gibney     U. Penn   A new candidate for the nef cone of M0,n 
23 Nov Tom Turkey     Plymouth Colony  ---------Thanksgiving Break-----------
 3 Dec: Dirk Kreimer IHES (France) Monday at 4:40! Hopf and Lie algebras for renormalizable quantum field theories
 7 Dec V. Retakh	 Rutgers   date(s) to change   TK

Fall Classes began September 4, 2007;  Final Exams began Friday, Dec 14, 2007.

Fall 2006/Spring 2007 Seminars

22 Sep: Corina Calinescu OSU     Intertwining vertex operators and combinatorial representation theory
 8 Dec* Haisheng Li    RU-Camden Certain generalizations of twisted affine Lie algebras and vertex algebras
30 Mar* Bill Cook      Rutgers   Vertex operator algebras and recurrence relations
 6 Apr* Antun Milas  SUNY-Albany  On a certain family of W-algebras
13 Apr* Vincent Graziano SUNY-Stony Brook  G-equvariant modular categories and Verlinde formula
20 Apr* Corina Calinescu OSU     Vertex-algebraic structure of certain modules for affine Lie algebras underlying recursions
27 Apr* Tom Robinson   Rutgers	 A Formal Variable Approach to Special Hyperbinomial Sequences

Fall Classes began September 5, 2006;  Final Exams began Friday, Dec 15
Spring 2007 Semester began Tuesday Jan 16; Spring Finals were May 3-9, 2007

Spring 2006 Seminars

20 Jan*: John Duncan    Yale  Vertex operators and sporadic groups
27 Jan(C) Jason Starr   MIT   Solutions of families of polynomial equations Colloquium at 4:00
 3 Feb  no seminar            (job interview talks)
10 Feb: Balazs Szegedy  IAS   Congruence subgroup growth of arithmetic groups in positive characteristic
17 Feb*: Haisheng Li  RU-Camden A smash product construction of nonlocal vertex algebras
24 Feb*: Andy Linshaw  Brandeis Chiral equivariant cohomology
 3 Mar: Wolmer Vasconcelos Rutgers "Complexity of the Normalization of Algebras"
10 Mar: Volodia Retakh     Rutgers "Algebras associated to directed graphs and related to factorizations of noncommutative polynomials"
17 Mar:  no seminar	-------------- Spring Break ------------- 
24 Mar   no seminar	----       D'Atri Lectures
31 Mar: Chuck Weibel      Rutgers  "Projective R[t]-modules and cdh cohomology"
 7 Apr  no seminars in April
 5 May Student Body Left  Rutgers   ---- Final Exam Grading Marathon -------

Classes begin January 18, 2006; Regular classes end Monday May 1. Final Exams are May 4-10, 2006.

Fall 2005 Seminars

During 2005-6, the Algebra Seminar (and Quantum Math Seminar) met on Fridays, at 1:00-2:00PM in H705.
  9 Sept Colonel Henry Rutgers -------- Department Reception ----------------
16 Sept: Thuy Pham        Rutgers   "jdeg of finitely generated graded algebras and modules"
   note room change to Hill 425 due to Kruskal Conference
23 Sept: Charles Weibel    Rutgers  "Effective Hodge structures"
30 Sept* Corina Calinescu  Rutgers  "On certain principal subspaces of standard modules and vertex operator algebras"
 7 Oct: Art DuPre    Rutgers-Newark "Extensions of Rings and their Endomorphisms"
14 Oct* Katrina Barron    Notre Dame "An isomorphism between two constructions of permutation-twisted modules for lattice vertex operator algebras"
21 Oct* Lin Zhang       RU+Sequent-Capital "Kazhdan-Lusztig's tensor category and the compatibility condition"
28 Oct: Bob Guralnick     USC & IAS "Rational Maps on the Generic Riemann Surface"
 4 Nov: Gene Abrams  U.Colorado/Colo.Springs "Leavitt path algebras"
11 Nov*  Siddhartha Sahi    Rutgers     "Supercategories and connections"
18 Nov: no seminar
25 Nov: Tom Turkey     Plymouth Colony  ---------Thanksgiving Break------------
 2 Dec: Earl Taft       Rutgers   "A class of left quantum groups: Variation on the theme of SL_q(n)"
 9 Dec: Harry Tamvakis  Brandeis  "Quantum cohomology of isotropic Grassmannians" (talk is at 12:30 in H423)
16 Dec*: Hisham Sati   U. Adelaide   "Mathematical aspects of the partition functions in string theory"
Semester begins Thursday September 1, 2005.
Regular classes end Tuesday, December 13. Final Exams are Dec.16-23.
Math Group Exam time is Friday Dec.16 (4-7PM)

From 1980 until Spring 2005, the seminar met on Fridays, at 2:50-4PM in H705 (Hill Center, Busch Campus).

Spring 2005 Seminars

21 Jan:    (first Friday of semester)  
28 Jan: no seminar       Job Interview Talks>
 4 Feb: Tom Graber       UC Berkeley   "Generalizations of Tsen's Theorem" (talk at 4:30 PM)
11 Feb: Pedro Barquero-Salavert CUNY Grad Center "Applications of the transfer method to quadratic forms and sheaves"
18 Feb: Christian Haesemeyer  IAS      "K-theory and cyclic homology of singularities"
25 Feb: Li Guo           RU-Newark     "Birkhoff decomposition in QFT and CBH formula"
 4 Mar: Earl Taft	 Rutgers       "Exotic Products of Linear Maps on Bialgebras"
11 Mar: Carlo Mazza	 IAS           "Schur Functors and Nilpotence Theorems"
18 Mar:  no seminar	-------------- Spring Break ------------- 
25 Mar: Zhaohu Nie    IAS/Stony Brook  "Karoubi's construction of Motivic Cohomology Operations"
 1 Apr: Gerhard Michler  U.Essen/Cornell "Uniqueness proof for Thompson's sporadic simple group"
 8 Apr:	Bin Shu         U.Virginia/E.Normal U.  "Representations and Forms of Classical Lie algebras over finite fields"
15 Apr: K. Ebrahimi-Fard Univ.Bonn     "Infinitesimal bialgebras and associative classical Yang-Baxter equations" 
21 Apr: Bruno Vallette    U.Nice        "Koszul duality" (Thursday at 1:10 p.m.)
22 Apr: Kate Hurley 
29 Apr: Cristiano Husu  U.Conn(Stamford) "Relative twisted vertex operators associated with the roots of the Lie algebras A_{1} and A_{2}"
 6 May: Student Body Left  Rutgers   ---- Final Exam Grading Marathon -------
7 June:	Miguel Ferrero  UF Rio Grande do Sol, Brazil "PARTIAL ACTIONS OF GROUPS ON ALGEBRAS" (talk at 4 PM)
Classes begin January 18, 2005
Spring Break is March 12-20, 2005
Regular classes end Monday May 2. Final Exams are May 5-11, 2005.

Fall 2004 Seminars

10 Sept Colonel Henry Rutgers -------- Department Reception ----------------
24 Sept*                         Yom Kippur is 9/25
 1 Oct* Liang Kong     Rutgers	 "Conformal field algebras and tensor categories"
 8 Oct:                          MacPherson's 60th Conference
15 Oct: Pavel Etingof  MIT	 "Cherednik and Hecke algebras of orbifolds"
22 Oct* Lin Zhang      RU+Sequent-Capital "When does the commutator formula imply the Jacobi identity in Vertex Operator Algebra theory?"
29 Oct*: A. Ocneanu    Penn State "Modular theory, quantum subgroups and quantum field theory"
 5 Nov: Helmut Hofer   Courant   D'Atri Lecture: Holomorphic Curve Methods (talk at 1:10 PM)
 5 Nov:* Keith Hubbard Notre Dame "Vertex Algebra coalgebras: Their operadic motivation and concrete constructions"
12 Nov: Chuck Weibel   Rutgers   "Homotopy theory for Motives"
19 Nov: Edwin Beggs  Univ. of Wales Swanswa "The Van Est spectral sequences for Hopf algebras"
26 Nov: Tom Turkey   Plymouth Colony ----------Thanksgiving Break------------
10 Dec: Edwin Beggs  Univ. of Wales Swanswa"Quasi-Hopf algebras, twisting and the KZ equation"
17 Dec: Student Body Left  Rutgers   ---- Final Exam Grading Marathon -------

Semester begins Wednesday September 1, 2003.
Regular classes end Monday, December 13. Final Exams are Dec.16-23.
Math Group Exam time is Thursday Dec.16 (4-7PM)

Spring 2004

26 Jan: Diane Maclagan   Stanford  "Toric Hilbert schemes" (talk at 4:30 PM)
28 Jan: Greg Smith       Columbia  "Orbifold Cohomology of Toric Stacks" (talk at 11:30 AM)
30 Jan: Anna Lachowska   MIT       "TBA" (talk at 1:10 PM)
 6 Feb: Chuck Weibel     Rutgers   "A survey of non-Desarguesian planes"
13 Feb: Kia Dalili       Rutgers   "The HomAB Problem"
20 Feb: Vladimir Retakh  Rutgers   "An Introduction to A-infinity Algebras"
27 Feb: Vladimir Retakh  Rutgers   "An Introduction to A-infinity Algebras II"
 5 Mar: Remi Kuku        IAS    "A complete formulation of the Baum-Connes Conjecture for the 
                                 action of discrete quantum groups"
12 Mar: Amnon Yekutieli  Ben Gurion Univ. "On Deformation Quantization in Algebraic Geometry"
19 Mar: no seminar	------------- Spring Break ------------- 
26 Mar: Alexander Retakh     MIT   "Conformal algebras and their representations"
 2 Apr: Aaron Lauve      Rutgers    "Capture the flag: towards a universal noncommutative flag variety"
 9 Apr* Stefano Capparelli Univ. Rome  "The affine algebra A22 and combinatorial identities"
16 Apr: Uwe Nagel       U.Kentucky  "Extremal simplicial polytopes"
16 Apr(C) Dale Cutkosky U. Missouri Colloquium Talk at 4:30 PM

23 Apr* Paul Rabinowitz Wisconsin    *** D'Atri Lecture at 1:10 PM ***
23 Apr: Li Guo          Rutgers-Newark "Dendriform algebras and linear operators"
30 Apr: Earl Taft         Rutgers   "There exists a one-sided quantum group"
 7 May Student Body Left  Rutgers   ---- Final Exam Grading Marathon --------

Classes begin January 20, 2004;   Spring Break is March 13-21, 2004
Regular classes end Monday May 3. Final Exams are May 6-12, 2004.
Math Group Final Exam time is Thursday May 6 (4-7PM)

Fall 2003 (in room H425)

 5 Sept George Willis   U. South Wales "scale functions on totally disconnected groups" 
 5 Sept Colonel Henry Rutgers -------- Department Reception ----------------
 8 Sept	Various people        -------- Gelfand 90th Birthday Celebration --------------
12 Sept Edwin Beggs     U.Wales-Swansea, UK  "Constructing tensor categories from from finite groups"
19 Sept Charlie Sims    Rutgers         "Algorithmic Questions in Rings of Rational Matrices?"
26 Sept David Radnell   Michigan       Thesis Defense:
       "Schiffer Variation in Teichmüller space, determinant line bundles and modular functors"
 3 Oct* Liang Kong	Rutgers        "Open-string vertex algebras"
10 Oct  C. Musili    U.Hyderabad, India "The Development of Standard Monomial Theory"
17 Oct Roy Joshua       Inst. Adv. Study "The Motivic DGA"
24 Oct Bodo Pareigis	Univ. Munich   "Modules, Comodules, Entwinings and Braidings"
31 Oct*	Benjamin Doyon  Rutgers		"From vertex operator algebras to the Bernoulli numbers"
 7 Nov* Geoffrey Buhl	Rutgers		"Complete reducibility and C_n-cofiniteness of vertex operator algebras"
14 Nov 	no RU seminar	------ Borel Memorial at IAS -----------
21 Nov* Lin Zhang       Rutgers		"A vertex operator algebra approach to the construction of a tensor category of Kazhdan-Lusztig"
28 Nov: Tom Turkey	----------Thanksgiving Break------------
 5 Dec* Victor Ostrik   IAS		"Finite extensions of vertex algebras"
12 Dec* Matt Szczesny   U. Penn.  	"Orbifolding the chiral de Rham complex"

Semester begins Tuesday September 2, 2003. Lewis Lectures are the week of October 3rd.
Regular classes end Wednesday, December 10. Final Exams are Dec. 15-22.
Math Group Exam time is Monday Dec.15 (4-7PM)

Spring 2003

28 Jan* Masahiko Miyamoto  Japan    "Interlocked modules and pseudo-trace functions"  
31 Jan:	no seminar	------------- Jean Taylor Symposium ------------- 
 5 Feb: Angela Gibney	Michigan "Some open questions about the geometry of the moduli space of curves"

21 Feb* Kiyokazu Nagatomo Japan  "Conformal field theory over the projective line"
28 Feb: Jooyoun Hong     Rutgers  "Normality of Rees algebras for conormal modules"
 7 Mar*: Yucai Su   Shanghai/Harvard    "Lie algebras associated with derivation-simple algebras"
14 Mar* Chengming Bai  Nankai&Rutgers "Novikov algebras and vertex (operator) algebras"
21 Mar: no seminar	------------- Spring Break ------------- 
28 Mar*: David Radnell   Rutgers   "Schiffer Variation in Teichmüller Space and Determinant Line Bundles"
 3 Apr: Claudio Pedrini U.Genova   "Finite dimensional motives"  Thursday 3PM - Note change in day!
 4 Apr# Hy Bass & Deborah Ball Michigan "Preparing teachers for the mathematical work of teaching"
11 Apr*: Lin Zhang       Rutgers   "Tensor category theory for modules for a vertex operator algebra -- introduction and generalization"
18 Apr: Constantin Teleman Cambridge U.  "Twisted K theory from the Dirac spectral flow"
25 Apr* Michael Roitman  Michigan    "Affinization of commutative algebras"
 2 May: Frederick Gardiner CUNY "The pure mapping class group of a Cantor set"  At 1:30 PM - Note change in time!
 9 May: Carlo Mazza      Rutgers   "Schur's Finiteness conditions in tensor categories" At 3:30 PM in H425 - Note change in time and room!

Regular classes end Monday May 5. Final Exams are May 8-14, 2003.
Math Group Final Exam time is Thursday May 8 (4-7PM)

Fall 2002

13 Sep: no seminar	Department Reception
20 Sep* YZ Huang        Rutgers  "Differential equations, duality and modular invariance"
27 Sep* Matthias Gaberdiel Kings College "Conformal field theory and vertex operator algebras"
 4 Oct: no seminar
11 Oct: Ravi Rao	TATA  "Raga Bhimpalasi: The Vaserstein-Suslin Jugalbandhi"
11 Oct(C) Igor Kriz	Michigan Colloquium Talk "Conformal field theory and elliptic cohomology" at 4:30 PM
18 Oct: Richard Stanley MIT 	Jacqueline Lewis Lecture at 4:30PM
18 Oct*: Earl Taft      Rutgers "Is there a one-sided quantum group?"
25 Oct:Christian Kassel CNRS-Univ. Louis Pasteur, Strasbourg "Explicit norm one elements for ring actions of finite abelian groups"
25 Oct(C) C. Kassel	""(Strasbourg) Colloquium Talk "Recent developments on Artin's braid groups" at 4:30PM
 1 Nov* Benjamin Doyon  RU Physics  "Twisted vertex operator algebra modules and Bernoulli polynomials"
 8 Nov: Charles Weibel  RU          "The work of Vladimir Voevodsky"
15 Nov* Takashi Kimura  IAS/Boston U. "Integrable systems and topology"
22 Nov: Julia Pevtsova  IAS   "Support Varieties for Finite Group Schemes"
29 Nov: Tom Turkey	----------Thanksgiving Break------------
 6 Dec: Anya Lachowska MIT "Modular group action in the center of the small quantum group"

Spring 2002

A 'j' marks a meeting of the Junior Algebra Seminar.
25 Jan: no seminar     Job Interview Talks
 1 Feb:	no seminar     Job Interview Talks
 8 Feb* Liz Jurisich  College of Charleston "The monster Lie algebra, Moonshine and generalized Kac-Moody algebras"
15 Feb:j Will Toler     RU Physics "Low dimensional topology and gauge theory"
22 Feb#  Laura Alcock   RU Math/Ed "The first course in real analysis in England: figuring out the conceptions students form"
 1 Mar:  -----		--		CANCELLED
 8 Mar*j Benjamin Doyon RU Physics "Vertex Operator Algebras and the Zeta function"
15 Mar*j Gordon Ritter  Harvard   "Montonen-Olive Duality in Yang-Mills Theory"
22 Mar: no seminar	------------- Spring Break ------------- 
29 Mar* Sergei Lukyanov RU Physics "Once again about Bethe Ansatz"
 5 Apr:j Benjamin Doyon RU Physics "Fractional Derivatives"
12 Apr: Lisa Carbone    RU "Lattice subgroups of Kac-Moody groups over finite fields"
19 Apr: Agata Smoktunowicz Yale/Warsaw(PAS) "A simple nil ring exists"
26 Apr: Earl Taft	RU  "Recursive Sequences and Combinatorial Identities"
 3 May* Yi-Zhi Huang	RU  "Differential equations and intertwining operators"
10 May: Calculus Profs	Rutgers	"Grading of Final Exams"

Regular classes end Monday, May 6. Final Exams end Wednesday, May 15.
Math Group Exam time is Thursday May 9th (4-7PM).

Fall 2001

 7 Sep:			Rutgers	Math Department Reception (4PM)
14 Sep* Sasha Kirillov	SUNY Stony Brook "On a q-analog of the McKay correspondence"
21 Sep: Ngo Viet Trung  Inst.Math.Hanoi "Hilbert functions of non-standard bigraded algebras"

 5 Oct: Ed Letzter	Temple  "Effective Representation Theory of Finitely Presented Algebras"
12 Oct* Yi-Zhi Huang	Rutgers  "Vertex operator algebras and conformal field theories"
19 Oct: V. Retakh	Rutgers  "Algebra and combinatorics of pseudo-roots of 
		             noncommutative polynomials and noncommutative differential polynomials" 
26 Oct*: Yan Soibelman	Kansas State U. "Elliptic curves and quantum tori"
 2 Nov* Yi-Zhi Huang	Rutgers  "Vertex operator algebras and conformal field theories II"
 9 Nov* Deepak Parashar MPI Leipzig "Some biparametric examples of Quantum Groups"
16 Nov* Yi-Zhi Huang	Rutgers  "Vertex operator algebras and conformal field theories III"
23 Nov: Tom Turkey	----------Thanksgiving Break------------
30 Nov* Hai-Sheng Li	Rutgers Camden "Certain noncommutative analogues of vertex algebras"
 7 Dec: Chuck Weibel	Rutgers	"Congruence subgroups of SL2(Z[1/n]), after Serre"
14 Dec:

regular classes end Wednesday, December 12. Final Exams are Dec. 15-22.
Math Group Exam time is Monday Dec.17 (4-7PM)

Spring 2001

26 Jan: Alexei Borodin	U.Penn ------- Job Candidate Interview -------
 2 Feb: Chuck Weibel	Rutgers	"POSTPONED TO March 30"
 9 Feb:	Dave Bayer	Columbia U. "Toric Syzygies and Graph Colorings"
16 Feb: Igor Kriz	U.Michigan  "A geometric approach to elliptic cohomology"
23 Feb* Yi-Zhi Huang	Rutgers	"Conformal-field-theoretic analogues of codes and lattices"
 2 Mar: Carl Futia	Southgate Capital Advisors "Bialgebras of Recursive Sequences and Combinatorial Identities"
 9 Mar* Haisheng Li	Rutgers Camden "Regular representations for vertex operator algebras"
16 Mar: no seminar	------------- Spring Break ------------- 
23 Mar* Yvan Saint-Aubin U.Montreal+IAS "Boundary behavior of the critical 2d Ising model"
30 Mar:	Chuck Weibel	Rutgers "Functors with transfer (on rings)"
 6 Apr*: Richard Ng	Towson U "The twisted quantum doubles of finite groups"
13 Apr*  Charles Doran	Columbia "Variation of the mirror map and algebra-geometric isomonodromic deformations"
20 Apr*: Lev Borisov	Columbia "Elliptic genera of singular algebraic varieties"
27 Apr: Diane Maclagan	IAS	"Supernormal vector configurations, Groebner fans, and the toric Hilbert scheme"
 4 May: Calculus Profs	Rutgers	"Grading of Final Exams"

Regular classes end Monday, April 30. Final Exams end Wednesday, May 9.
Math Group Exam time is Thursday May 3rd (4-7PM)

Fall 2000

 8 Sep: Amelia Taylor	Rutgers	"The inverse Gröbner basis problem in codimension two"
15 Sep* Mike Douglas	RU Physics "D-branes"
22 Sep: Chuck Weibel	Rutgers	"Topological vs. algebraic $K$-theory for complex varieties"
29 Sep: no seminar	------------- Rosh Hoshanna ------------
 6 Oct: Daya-Nand Verma	TATA Inst. "Progress Report on the Jacobian Conjecture"
13 Oct: no seminar
20 Oct* Constantin Teleman U.Texas "The Verlinde algebra and twisted K-theory"
27 Oct: Chuck Weibel	Rutgers	"Homotopy Ends and Thomason Model Categories"
 3 Nov* Mirko Primc	U.Zagreb  "Annihilating fields of standard modules of sl_2~ and combinatorial identities"
10 Nov: Suemi Rodriguez-Romo UNAM Mexico "Quantum Group Actions on Clifford Algebras"
17 Nov: Craig Huneke	U.of Kansas "Growth of Symbolic Powers in Regular Local Rings"
24 Nov: Tom Turkey	----------Thanksgiving Break------------
 1 Dec# Nina Fefferman and Matt Young  Rutgers  VIGRE presentations on p-adic numbers 
 8 Dec* Mike Douglas?	RU Physics  "D-branes, instantons and orbifolds"

Winter 2000

 4 Feb: Martin Sombra	IAS+LaPlata "Division formulas and the arithmetic Nullstellensatz"
11 Feb: no seminar
18 Feb: Claudio Pedrini IAS+Genoa "K-theory of algebraic varieties: a Survey"
25 Feb: M.R.Kantorovitz IAS	"Andre-Quillen homology from a calculus viewpoint"
				(with Hochschild homology and algebraic K-theory for dessert)
 3 Mar: S. Hildebrandt	Bonn	*** D'Atri Lecture *** (2-dim. Variational Problems)
10 Mar: D. Christensen  IAS	"Brown representability in derived categories"
17 Mar:		---	----	-------	Spring Break -----------
24 Mar* Haisheng Li	RU-Camden "Certain extended vertex operator algebras"
31 Mar* Christoph Schweigert Paris "Conformal boundary conditions and three-dimensional topological field theory"
 7 Apr: no seminar
14 Apr* Christian Schubert LAPTH France "Multiple Zeta Value Identities from Feynman Diagrams"
21 Apr: no seminar
28 Apr* Tony Milas	Rutgers	"Structure of fusion rings associated to Virasoro vertex operator algebras"
 3 May* (Wednesday) Tony Milas Rutgers	"Differential operators and correlation functions"

Fall 1999

24 Sep: V. Retakh	Rutgers "Noncommutative rational functions+Farber's invariants of boundary links"
 1 Oct: Antun Milas*	Rutgers	"Intertwining operator superalgebras for N=1 minimal models"
 8 Oct: Fedor Bogomolov	NY Univ	"Fundamental Groups of Projective Varieties"
15 Oct: Earl Taft	Rutgers	"Sequences satisfying a polynomial recurrence"
22 Oct: Yuji Shimizu*	Kyoto U "Momentum mappings and conformal fields"
29 Oct: Leon Seitelman  U.Conn.  SPECIAL VIGRE LECTURE
		"What's a mathematician like you doing in a place like that"
 5 Nov:	Keith Pardue	IDA/Princeton  "Generic Polynomials"
12 Nov: *Haisheng Li  Rutgers (Camden) "The Diamond lemma for algebras (following Bergman)"
19 Nov: Yuri Tschinkel	U.Illinois "Equivariant compactifications of G_a^n"
26 Nov: Tom Turkey	------Thanksgiving Break--------
 3 Dec: Borisov*	Columbia "Vertex algebras and mirror symmetry"
10 Dec: Chongying Dong	UC Santa Cruz "Holomorphic orbifold theory, quantum doubles and dual pairs" 

Spring 1999

22 Jan:	P. Balmer	Rutgers	"The derived Witt group of a ring"
29 Jan: W. Vasconcelos	Rutgers	"The intertwining algebra"
 5 Feb: Thomas Geisser	U.Tokyo	"TBA"
12 Feb:Dennis Gaitsgory Harvard/IAS "On a VOA of differential operators on a loop group"
19 Feb: Mark Walker	Nebraska "The total Chern class map"
26 Feb: Michael Roitman Yale	"Universal constructions in conformal and vertex algebras"
 5 March: E. Friedlander Northwestern "Re-interpreting the Bloch-Lictenbaum spectral sequence"
12 March: R. Schoen		D'Atri Lecture instead of seminar
19 March: Vernal Equinox	------Spring Break March 14-21----
26 March: Yuji Shimizu	Kyoto and Rutgers "Conformal blocks and KZB equations"
 2 April: Roger Rabbit	Toontown	no seminar (Passover/Easter)
 9 April: 
16 April: Marco Schlichting RU and U. Paris "The negative K-theory of an exact category"
23 April: Chuck Weibel	Rutgers	"Projective modules over normal surfaces"
30 April: Percy Deift	Courant Institute (Colloquium talk)
 7 May:  Yuji Shimizu	Kyoto and Rutgers "Geometric structures underlying some conformal field theories"

Fall 1998

18 Sep: Lowell Abrams	Rutgers		"Modules, comudules and cotensor products over Frobenius algebras"
25 Sep: Bogdan Ion	Princeton	"Maschke's theorem revisited"
 2 Oct: Haisheng Li(*)	Rutgers Camden	"An infinite-dimensional analogue of Burnside's theorem"
 9 Oct: Aron Simis	Univ.F.Pernambuco (Recife, Brazil) "Geometric Aspects of Rees Algebras"
16 Oct: A. Beilinson	Univ. Chicago	Colloquium in honor of Gelfand
23 Oct: Michael Finkelberg(*) IAS/Independent Moscow Univ. "An integrable system on the space of based maps from P^1 to a flag variety"
30 Oct: Yi-Zhi Huang(*)	Rutgers		"Semi-infinite forms and topological vertex operator algebras"
 6 Nov:	Alfons Ooms	Limburgs Univ, Belgium	"On the Gelfand-Kirillov conjecture"
13 Nov: A. Kirillov, Jr.(*) IAS		"On the Lego-Teichmuller game"
20 Nov:	M.F. Yousif	Ohio State-Lima	"On three conjectures on quasi-Frobenius Rings"
27 Nov:	Tom Turkey	------Thanksgiving Break--------
 4 Dec: C. Lenart	Max Planck (Bonn)	""
11 Dec: S. Majid	Cambridge Univ.	"braided groups and the inductive construction of U_q(g)"

Spring 1998

30 Jan: C. Weibel	Rutgers		"local homology vs. cohomology (after Greenlees-May)"
 6 Feb: Brian Parshall	U. of Virginia	"The cohomology and representation theory of reductive groups in non-describing characteristics"
13 Feb: M. Khovanov(*)	Yale and IAS	"Lifting the Jones polynomial of knots to invariants of surfaces in 4-space"
20 Feb: Ming-Sun Li	Rowan Univ.	"Spectral matrices associated to an algebra"
27 Feb: Yi-Zhi Huang(*)	Rutgers		"Analytic aspects of Intertwining Operators"
 6 Mar:	Boris Khesin(*)	IAS+U.Toronto   "Geometric complexification of affine algebras and flat connections on surfaces"
13 Mar: no algebra seminar
20 Mar:	Vernal	Equinox		------Spring Break--------
27 Mar: N. Inassaridze	Razmadze Inst.	"Non-abelian homology of groups"
 3 Apr: Jim Stasheff	UNCarolina	"Physically inspired homological algebra"
10 Apr: Movshev(*)	...	QUANTUM MATH SEMINAR
17 Apr: S. Sahi		Rutgers		"A new character formula for compact Lie groups"
24 Apr: Stefan Schmidt	Berkeley	"Projective Geometry of Modules"

Fall 1997

19 Sep: Bill Kantor	U. Oregon  Colloquium: "Black box classical groups"
26 Sep: Lowell Abrams	Rutgers	  "2-dimensional TQFT's and Frobenius Algebras"
 3 Oct:	---		------ Rosh Hoshanna -----
10 Oct: Tor Gunston	Rutgers		"Degree functions and linear resolutions"
31 Oct: Chuck Weibel	Rutgers		"introducing Motives"
 7 Nov: ---		Columbia Univ.	Bass Conference
14 Nov: Stefan Catoiu	Temple Univ.	"IDEALS OF THE ENVELOPING ALGEBRA U(sl_2)"
21 Nov: M. Kontsevich	IHES		"Deformation, Quantization and Beyond"
28 Nov:	Tom Turkey	------Thanksgiving Break--------
 5 Dec: M. Kontsevich	IHES		"Deformation, Quantization and Beyond"
12 Dec: C. Pedrini	U. Genova	"K-Theory and Bloch's Conjecture for complex surfaces" 

Spring 1997

31 Jan: Luisa Doering  	Rutgers      "Generalized Hilbert functions"
 7 Feb: 	postponed
14 Feb:	Miguel Ferrero	Porto Alegre,Brazil "Closed and prime submodules of centered bimodules and applications to ring extensions"
21 Feb: Richard Ng	Rutgers	     "Freeness of Hopf algebras over subalgebras"
28 Feb: Siddartha Sahi	Rutgers	     "Introduction to Macdonald polynomials"
 7 Mar: Barbara Osofsky	Rutgers	     "Projective dimension for commutative von Neumann regular rings and a new lattice invariant"
14 Mar: Chuck Weibel	Rutgers	     "K-theory and zeta functions on number fields"
21 Mar:	------------	Spring Break  ------------
28 Mar: Carl Faith	Rutgers      "Rings with ACCs on annihilators"
 4 Apr: Joe Brennan	N.Dakota     "The Ends of Ideals"
11 Apr: Jan Soibelman	Kansas State "Meromorphic tensor categories and quantum affine algebras"
18 Apr: Chuck Weibel	Rutgers	     "Tor without identity (after Quillen)"
25 Apr: Wolmer Vasconcelos Rutgers   "Integral closure"
 2 May: Luca Mauri	Rutgers	     "2 torsors" 

Fall 1996

20 Sep: C. Weibel    	Rutgers	"the 2-torsion in the K-theory of Z"
27 Sep: Tor Gunston	Rutgers "Cohomological dimension of graded modules"
 4 Oct:	B. Ulrich     MichState "Divisor class groups and Linkage"
11 Oct:  --		IAS	Langlands Fest
18 Oct:	Bob Guralnick	USC	"Finite Orbit Modules and Double Cosets for Algebraic Groups"
25 Oct: Richard Weiss	Tufts	"Moufang polygons"
 1 Nov: Georgia Benkart Wisconsin "Lie Algebras Graded by Finite Root Systems"
 8 Nov: Richard Ng	Rutgers "On the projectivity of module coalgebras"
15 Nov:  --			no seminar
22 Nov: Bill Keigher	RU-Newark "The ring of Hurwitz series"
29 Nov: Tom Turkey 	Thanksgiving		(no seminar)
 6 Dec: Leon Pritchard	RU-Newark "Hurwitz series Formal Functions"
13 Dec: Reading Period after classes 

Spring 1996

26 Jan: A.Corso    Rutgers	"generic gaussian ideals"
 2 Feb: no seminar
 9 Feb: E. Taft	   Rutgers	"Quantum Convolution"
16 Feb: Frosty S.  Weather	"Snow storm--talks rescheduled"
23 Feb: B. Leasher Rutgers	"Geometric Aspects of Steinberg Groups for Jordan Pairs"
 1 Mar: L. Mauri   Rutgers	"Low-dimensional Descent theory"
 8 Mar: K.Consani  IAS		"Double complexes and local Euler factors on algebraic degeneration"
15 Mar: ------------	Spring Break  ------------
22 Mar: YZ Huang   Rutgers	"On algebraic D-modules and vertex algebras"
29 Mar: Doering&Gunston Rutgers "Algebras Arising from Bipartite Planar Graphs"
 5 Apr: Consuelo Martinez  Yale "Power subgroups of profinite groups"
12 Apr: M. Singer  NC State	"Galois theory for difference equations"
19 Apr: C. Weibel  Rutgers	"Popescu Desingularization (after Swan)"
26 Apr: R. Hoobler CCNY		"Merkuriev-Suslin Theorem for arbitrary semi-local rings"
14 May: K. Mimachi Kyushu U.	"Quantum Knizhink-Zamolodchikov equation and eigenvalue problem of Macdonald equations" 

Fall 1995

28 Sep: M.Gerstenhaber U. Penn  "Symplectic structures on max. parabolic
				subgps. of SL_n and boundary solutions of
				the classical Yang-Baxter equation"
29 Sept:W. Vasconcelos Rutgers	"Gauss Lemma"
 6 Oct: I. Gelfand     Rutgers	"Noncommutative symmetric functions"
13 Oct: Joan Elias     Barcelona"On the classification of curve singularities"
20 Oct: B. Osofsky     Rutgers  "Connections between foundations and Algebra"
27 Oct: O. Stoyanov    Rutgers	"Quantum Unipotent Groups"
 3 Nov: I. Gelfand     Rutgers  "Noncommutative Grassmannians"
10 Nov: M. Tretkoff    Stevens  "Rohrlich's formula for hypersurface periods"
17 Nov: C. Weibel      Rutgers  "Tinker Toys and graded modules"
24 Nov: Tom Turkey		Thanksgiving Break
 1 Dec: Siu-Hung Ng    Rutgers  "Lie bialgebra structures on the Witt algebra"
 8 Dec: E. Zelmanov    Yale	"On narrow groups and Lie algebras"

Spring 1995

27 Jan	Alberto Corso	Rutgers	"Links of irreducible varieties"
 3 Feb	Chuck Weibel	Rutgers "Operads for the Working Mathematician"
10 Feb	Maria Vaz Pinto Rutgers "Hilbert Functions and Sally Modules"
17 Feb	Yi-Zhi Huang	Rutgers	"Vertex Operator Algebras for Lay Algebraists"
24 Feb	O. Matthieu	"On the modular representations of the symmetric group"
 3 Mar	Claudio Pedrini Genova	"The Chow group of singular complex surfaces"
10 Mar	B.Sturmfels-Berkly A normal form algorithm for modules over k[x,y]/(xy)
18 Mar	------------	Spring Break  ------------
24 Mar	Francesco Brenti IAS "Twisted incidence algebras and 
					Kazhdan-Lusztig-Stanley functions"
31 Mar	Myles Tierney Rutgers 	"Simplicial sheaves"
 7 April Wolmer Vasconcelos	"A Lemma of Gauss"
14 April Peter Cottontail	"Easter's on its way!  (Passover too!)"
21 April Susan Morey  "Symbolic Powers, Serre Conditions and CM Rees algebras"
28 April K.P. Shum Hong Kong/Maryland "Regular semigroups and generalizations"


29 Jan: Earl Taft     Rutgers   "Linearly recursive sequences in several variables"
 5 Feb: J. Brennan  N.D. State  "Integral closure of a morphism"
12 Feb: Chuck Weibel  Rutgers   "Chern classes and torsion in algebraic K-theory"
19 Feb: Friedrich Knop Rutgers  "Invariant valuations and eqeuivariant embeddings"
26 Feb: Charles Walters Rutgers "Projectively normal curves"
 4 Mar: M.C. Kang     Taiwan    "Monomial group actions on rational functino fields"
11 Mar: Art Dupre  Rutgers-Newark "Extensions and cohomology of groups"
27 Mar: Wolmer Vasconcelos Rutgers "The top of a system of equations"
 3 Apr: Marvin Tretkoff  Stevens "Some of Dwork's Cohomology Spaces"
17 Apr: Frederico Bien Princeton "Vanishing theorems for D-modules on spherical varieties"
24 Apr: Carl Faith  Rutgers     "FPF rings"


13 Sep: W. Vogell Martin-Luther U. "Intersection theory"
25 Sep: Sunsook Noh    Rutgers  "Divisors of 2nd kind and 2-dimensional regular local rings"
 2 Oct: W. Vasconcelos Rutgers  "The Sally module of a reduction"
 9 Oct: Earl Taft      Rutgers  "Witt and Virasoro algebras as Lie bialgebras"
16 Oct: Bill Hoyt      Rutgers  TBA
23 Oct: Barbara Osofsky Rutgers TBA
30 Oct: V. Retakh      Rutgers  "The algebra of extensions without resolutions"
 6 Nov: V. Retakh      Rutgers  continued
13 Nov: A. Brownstein          "Generalized Braid groups and motions of strings in 3-space"
20 Nov: Aron Simis U.Fed.Bahia  "On tangent cones"


25 Jan Chuck Weibel    Rutgers  "Operations and symbols in K-theory"
 1 Feb Barr Von Oehsen Rutgers  "Elliptic genera and Jacobi polynomials"
 8 Feb Bernie Johnston FAU      TBA
22 Feb Barbara Osofsky Rutgers  "Constructing nonstandard uniserial modules over valuation domains"
 1 Mar W. Vasconcelos  Rutgers  "Explicit Nullsatzen"
 8 Mar  Matt Miller  S. Carolina "Betti numbers of modules of finite length"
15 Mar  Sam Vovsi    Ryder College TBA
21 Mar R.I. Grigorchuk Moscow Inst.  "The Burnside problem"
 5 Apr Rafael Villareal Rutgers  TBA
12 Apr Willie Cortinas Buenos Aires TBA
19 Apr Joachim Lambek  McGill U. TBA
26 Apr Charles Walter  Rutgers   "Algebraic space curves with the expected monomial"
 3 May Bernd Ulrich Michigan State "Projective curves and their hypersurface sections"


15 Sep Hisao Tominaga  Okayama U. "Some polynomial identites and commutativity of rings"
22 Sep Chuck Weibel   Rutgers  "Set-theoretic complete intersection points on curves"
 6 Oct Carl Faith      Rutgers "Subrings of FPF and self-injective rings"
13 Oct W. Vasconcelos  Rutgers "Symmetric algebras and syzygies"
20 Oct David Rohrlich  Rutgers "maps to the projective line of minimal degree"
27 Oct Stan Page  U.Br.Columbia "Slim rings and modules"
 3 Nov Joe Johnson     Rutgers  "Dimension fans and finite presentation of graded modules"
10 Nov Ron Donagi      Utah     "The Schottky problem"
17 Nov J. Dorfmesiter  Muenster "Siegel domains"
24 Nov Barbara Osofsky Rutgers  "Strange self-injective rings"
 1 Dec Bill Hoyt       Rutgers  "Division points on generic elliptic curves"
 8 Dec Chuck Weibel    Rutgers  "KABI"


28 Jan S. Goto  Brandeis/Nihon U.  "On Buchsbaum rings"
 4 Feb Chuck Weibel   Rutgers  "When are projective modules extended?"
11 Feb Barbara Osofsky Rutgers "Between flatness and projectivity"
18 Feb David Rohrlich Rutgers  "An intro to L-functions on elliptic curves"
25 Feb M. Takeuchi    IAS      "Commutative Hopf algebras and cocommutative Hopf algebras in char. p"
04 Mar Harry Gonshor  Rutgers  "Conway numbers and semigroup rings"
11 Mar Louis Rowen    Yale     "Finite dimensional division algebras"
25 Mar Bill Hoyt      Rutgers  "Periods of abelian integrals"
 1 Apr Don Schack   U. Buffalo "Deformations of diagrams"
 8 Apr Earl Taft      Rutgers  "Hopf algebras"
15 Apr Chuck Weibel   Rutgers  "Witt vectors made easy"
22 Apr Jan van Geel U. Antwerp "Primes and value functions"
29 Apr Dick Cohn      Rutgers  "The General Solution in Differential Equations"


25 Sept Earl Taft     Rutgers  "A generalization of divided power sequences"
 1 Oct  Moss Sweedler Cornell  "Products of flat modules"
 8 Oct  Chuck Weibel  Rutgers  "Principal ideals and smooth curves"
15 Oct  Joe Johnson   Rutgers  "Rings that lack ..."
22 Oct  Wolmer Vasconcelos Rutgers  TBA
29 Oct Richard Block UC Riverside "Irreducuble representations of skew polynomial rings"
 5 Nov  M. Gerstenhaber U.Penn  "On the deformation of differential graded algebras"
 7 Nov  F. Orecchia   U.Genoa   "Tangent cones and singularities of algebraic curves"
12 Nov  E. Sontag     Rutgers   "PL Algebras"
19 Nov  Dick Bumby    Rutgers   "Jacobi symbols"
 3 Dec  Carl Faith    Rutgers   "Noncommutative rings"

Abstracts of seminar talks

Spring 2018

Singularities in geometry, topology and strings, Antonella Grassi, May 9, 2018):
I will discuss a "Brieskorn-Grothendieck" program involving certain singularities, Lie algebras and representations. These singularities arise in many different areas of mathematics and physics. I will focus on the case of complex 3 dimensional spaces relating to algebraic geometry, topology and physics. I will disucss local, global and local-to-global properties of threefolds with certain singularities and crieteria for these threefolds to be rational homology manifolds and conditions for threefolds to satisfy rational Poincar\'e duality.
We state a conjecture on the extension of Kodaira's classification of singular fibers on relatively minimal elliptic surfaces to the class of birationally equivalent relatively minimal genus one fibered varieties and we give results in this direction.

Hodge theory and o-minimal geometry (Benjamin Bakker, May 2, 2018):
Hodge structures on cohomology groups are fundamental invariants of algebraic varieties; they are parametrized by quotients $D/\Gamma$ of periods domains by arithmetic groups. Except for a few very special cases, such quotients are never algebraic varieties, and this leads to many difficulties in the general theory. We explain how to partially remedy this situation by equipping $D/\Gamma$ with an o-minimal structure, and show that period maps are "definable" with respect to this structure.
As a consequence, we obtain an easy proof of a result of Cattani--Deligne--Kaplan on the algebraicity of Hodge loci, a strong piece of evidence for the Hodge conjecture. The proof of the main theorem relies heavily on work of Schmid, Kashiwara, and Cattani--Kaplan--Schmid on the asymptotics of degenerations of Hodge structures. This is joint work with B. Klingler and J. Tsimerman.

Singularities of general fibers in positive characteristic (Joe Waldron, April 18, 2018):
Generic smoothness fails to hold for some fibrations in positive characteristic. We study consequences of this failure, in particularly by obtaining a canonical bundle formula relating a fiber with the normalization of its maximal reduced subscheme. This has geometric consequences, including that generic smoothness holds on terminal Mori fiber spaces of relative dimension two in characteristic $p\geq 11$. This is joint work with Zsolt Patakfalvi.

Conformal blocks attached to twisted groups (Chiara Damiolini, April 11, 2018):
Let $G$ be a simple and simply connected algebraic group over $\mathbb{C}$. We can attach to $G$ the sheaf of conformal blocks: a vector bundle on $M_{g}$ whose fibres are identified with global sections of a certain line bundle on the stack of $G$-torsors. We generalize the construction of conformal blocks to the case in which $\mathcal{G}$ is a twisted group over a curve which can be defined in terms of covering data. In this case the associated conformal blocks define a sheaf on a Hurwitz space and have properties analogous to the classical case.

Derived categories of moduli spaces of stable rational curves (Ana-Maria Castravet, March 28, 2018):
A question of Manin is whether the derived category of the Grothendieck-Knudsen moduli space $M_{0,n}$ of stable, rational curves with n markings admits a full, strong, exceptional collection that is invariant under the action of the symmetric group $S_n$. I will present an approach towards answering this question. In particular, I will explain a construction of an invariant full exceptional collection on the Losev-Manin space. This is joint work with Jenia Tevelev.

The indeterminacy of universal Abel-Jacobi sections (Nicola Pagani, March 14, 2018):
The (universal) Abel-Jacobi maps are the sections of the forgetful morphism from the universal Jacobian to the corresponding moduli space $M_{g,n}$ of smooth pointed curves. When the source and target moduli spaces are compactified, these sections are only rational maps, and it is natural to ask for the largest locus where each of them is a well-defined morphism. We explicitly characterize this locus, which depends on the chosen compactification of the universal Jacobian (for the source we fix the Deligne-Mumford compactification $\bar{M}_{g,n}$ by means of stable curves). In particular, we deduce that for every Abel-Jacobi map there exists a compactification of the universal Jacobian such that the map extends to a well-defined morphism on $\bar{M}_{g,n}$. We apply this to the problem of defining and computing several different extensions to $\bar{M}_{g,n}$ of the double ramification cycle (= the locus of smooth pointed curves that admit a meromorphic function with prescribed zeroes and poles at the points).
This is a joint work with Jesse Kass.

Algebraic dynamics from topological and holomorphic dynamics (Rohini Ramadas, Feb. 28, 2018):
Let $f:S^2 \to S^2$ be an orientation-preserving branched covering from the 2-sphere to itself whose postcritical set $P := \{ f^n(x) | x\ \mathrm{is\ a\ critical\ point\ of\ f\ and}\ n>0 \}$ is finite. Thurston studied the dynamics of $f$ using an induced holomorphic self-map $T(f)$ of the Teichmuller space of complex structures on $(S^2, P)$. Koch found that this holomorphic dynamical system on Teichmuller space descends to algebraic dynamical systems:
1. $T(f)$ always descends to a multivalued self map $H(f)$ of the moduli space $M_{0,P}$ of markings of the Riemann sphere by the finite set $P$
2. When $P$ contains a point $x$ at which $f$ is fully ramified, under certain combinatorial conditions on $f$, the inverse of $T(f)$ descends to a rational self-map $M(f)$ of projective space $P^n$. When, in addition, $x$ is a fixed point of $f$, i.e. $f$ is a `topological polynomial', the induced self-map $M(f)$ is regular.
The dynamics of $H(f)$ and $M(f)$ may be studied via numerical invariants called dynamical degrees: the k-th dynamical degree of an algebraic dynamical system measures the asymptotic growth rate, under iteration, of the degrees of k-dimensional subvarieties.
I will introduce the dynamical systems $T(f)$, $H(f)$ and $M(f)$, and dynamical degrees. I will then discuss why it is useful to study $H(f)$ (resp. $M(f)$) simultaneously on several compactifications of $M_{0,P}$. We find that the dynamical degrees of $H(f)$ (resp. $M(f)$) are algebraic integers whose properties are constrained by the dynamics of $f$ on the finite set $P$. In particular, when $M(f)$ exists, then the more $f$ resembles a topological polynomial, the more $M(f): P^n \to P^n$ behaves like a regular map.

Curves, maps, and singularities in genus one (Dhruv Ranganathan, February 21, 2018):
I will outline a new framework based on tropical and logarithmic methods to study genus one curve singularities and discuss its relationship with the geometry of moduli spaces. I will focus on two applications of these ideas.
First, they allow one to explicitly factorize the rational maps among log canonical models of the moduli space of n-pointed elliptic curves. Second, they reveal a modular interpretation for Vakil and Zinger's famous desingularization of the space of elliptic curves in projective space, a short conceptual proof of that result, and several new generalizations.
Time permitting, though it rarely does, I will mention some applications to both classical and virtual enumerative geometry. This is based on work with Len and with Santos-Parker and Wise, as well as ongoing work with Battistella and Nabijou.

Equations of Cartwright-Steger surface (Lev Borisov, February 7, 2018):
Cartwright-Steger surface is an algebraic surface of general type which appeared in the study of fake projective planes. I will describe the technique that allowed us to find equations of it, in its bicanonical embedding. This is a joint work with Sai Kee Yeung.

An arithmetic site of Connes-Consani type for Gaussian integers (Aurélien Sagnier, Jan. 24, 2018):
Connes and Consani proposed to study the action of the multiplicative monoid of positive integers $\mathbb{N}^\times$ on the tropical semiring $(\mathbb{Z},max,+)$, as an approach to the Riemann zeta function. This construction depends upon the ordering on the reals. I will first explain their approach, then give an extension of this construction to the Gaussian integers.

Fall 2017

Gonality of complete intersection curves (Brooke Ullery, Dec. 13, 2017:
The gonality of a smooth projective curve is the smallest degree of a map from the curve to the projective line. If a curve is embedded in projective space, it is natural to ask whether the gonality is related to the embedding.
In my talk, I will discuss recent work with James Hotchkiss. Our main result is that, under mild degree hypotheses, the gonality of a complete intersection curve in projective space is computed by projection from a codimension 2 linear space, and any minimal degree branched covering of $\mathbb P^1$ arises in this way.

Equivariant K-theory associated to Kac-Moody groups (Seth Baldwin, Dec. 6, 2017):
The cohomology ring of flag varieties has long been known to exhibit positivity properties. One such property is that the structure constants of the Schubert basis with respect to the cup product are non-negative. Brion (2002) and Anderson-Griffeth-Miller (2011) have shown that positivity extends to K-theory and T-equivariant K-theory, respectively. In this talk I will discuss recent work (joint with Shrawan Kumar) which generalizes these results to the case of Kac-Moody groups.

Local-global principles for rational points and zero-cycles (Julia Hartmann, Nov. 15, 2017):
Given a variety over a field $F$ and a collection of overfields of $F$, one may ask whether the existence of rational points over each of the overfields (local) implies the existence of a rational point over $F$ (global). Such local-global pinciples are a main tool for understanding the existence of rational points on varieties.
     In this talk, we study varieties that are defined over semi-global fields, i.e., function fields of curves over a complete discretely valued field. A semi-global field admits several natural collections of overfields which are geometrically motivated, and one may ask for local-global principles with respect to each such collection. We exhibit certain cases in which local-global principles for rational points hold. We also show that local-global principles for zero-cycles of degree one hold provided that local-global principles hold for the existence of rational points over extensions of the function field. This last assertion is analogous to a known result for varieties over number fields.
(Joint work with J.-L. Colliot-Thélène, D. Harbater, D. Krashen, R. Parimala, and V. Suresh)

Explicit equations of a fake projective plane (Lev Borisov, Nov. 8, 2017):
Fake projective planes are complex algebraic surfaces of general type whose Betti numbers are the same as that of a usual projective plane. The first example was constructed by Mumford about 40 years ago by 2-adic uniformization. There are 50 complex conjugate pairs of such surfaces, given explicitly as ball quotients (Cartwright+Steger). However, a ball quotient description does not on its own lead to an explicit projective embedding. In a joint work with JongHae Keum, we find equations of one pair of fake projective planes in bicanonical embedding, which is so far the only result of this kind.

Birational geometry of moduli spaces of parabolic bundles (Han-Bom Moon, October 18, 2017):
I will describe a project on birational geometry of the moduli space of parabolic bundles on the projective line in the framework of Mori's program, and its connection with classical invariant theory and conformal blocks. This is joint work with Sang-Bum Yoo.

Linear Systems on General Curves of Fixed Gonality (David Jensen, Oct. 4, 2017:
The geometry of an algebraic curve is governed by its linear systems. While many curves exhibit bizarre and pathological linear systems, the general curve does not. This is a consequence of the Brill-Noether theorem, which says that the space of linear systems of given degree and rank on a general curve has dimension equal to its expected dimension. In this talk, we will discuss a generalization of this theorem to general curves of fixed gonality. To prove this result, we use tropical and combinatorial methods.
This is joint work with Dhruv Ranganathan, based on prior work of Nathan Pflueger.

Character levels and character bounds (Pham Huu Tiep, September 27, 2017):
We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give various characterizations of the level of a character in terms of its Lusztig's label, of its degree, and of certain dual pairs. This concept is then used to prove exponential bounds for character values, provided that either the level of the character or the centralizer of the element is not too large. This is joint work with R. M. Guralnick and M. Larsen.

K-classes of Brill-Noether loci and a determinantal formula (Nicola Tarasca, September 20, 2017):
I will present a formula for the Euler characteristic of the structure sheaf of Brill-Noether loci of linear series on curves with prescribed vanishing at two marked points.
The formula recovers the classical Castelnuovo number in the zero-dimensional case, and previous work of Eisenbud-Harris, Pirola, Chan-López-Pflueger-Teixidor in the one-dimensional case. The result follows from a new determinantal formula for the K-theory class of certain degeneracy loci of maps of flag bundles.
This is joint work with Dave Anderson and Linda Chen.

A general algebraic structure theory for tropical mathematics (Louis Rowen, September 13, 2017):
We study triples (A,T,-) of a set A with algebraic structure (such as a semiring), a subset T and a negation operator '-' on T. A key example is the max-plus algebra T. This viewpoint enables one to view the tropicalization functor as a morphism, suggesting tropical analogs of classical structures such as Grassmann algebras, Lie algebras, Lie superalgebras, Poisson algebras, and Hopf algebras.


Spring 2017

Equivariant quantum K-theory of projective space (Sjuvon Chung, April 26, 2017):
Recent developments of Buch-Chaput-Mihalcea-Perrin have allowed for a closer look at the quantum K-theory of cominuscule flag varieties. For example, their Chevalley formula allows one to compute quantum K-theoretic products involving Schubert divisor classes. In the special case of projective space, one can extend this Chevalley formula to describe products of arbitrary Schubert classes. We shall discuss this extension along with some of its potential combinatorial and representation-theoretic consequences.

Schubert calculus (Anders Buch, April 19, 2017):
The cohomology ring of a flag variety has a natural basis of Schubert classes. The multiplicative structure constants with respect to this basis count solutions to enumerative geometric problems; in particular they are non-negative. For example, the structure constants of a Grassmannian are the classical Littlewood-Richardson coefficients, which show up in numerous branches of mathematics.
I will speak about a new puzzle-counting formula for the structure constants of 3-step partial flag varieties that describes products of classes that are pulled back from 2-step flag varieties. By using a relation between quantum cohomology of Grassmannians and classical cohomology of 2-step flag varieties, this produces a new combinatorial formula for the (3 point, genus zero) Gromov-Witten invariants of Grassmannians, which is in some sense more economical than earlier formulas.

Kirillov-Reshetikhin modules and Macdonald polynomials: a survey and applications (Christian Lenart, April 5, 2017):
This talk is largely self-contained.
In a series of papers with S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, we developed a uniform combinatorial model for (tensor products of one-column) Kirillov-Reshetikhin (KR) modules of affine Lie algebras. We also showed that their graded characters coincide with the specialization of symmetric Macdonald polynomials at $t=0$, and extended this result to non-symmetric Macdonald polynomials. I will present a survey of this work and of the recent applications, which include: computations related to KR crystals, crystal bases of level 0 extremal weight modules, Weyl modules (local, global, and generalized), $q$-Whittaker functions, and the quantum $K$-theory of flag varieties.

Dynamics and polynomial invariants for free-by-cyclic groups (Ilya Kapovich, March 22, 2017):
We develop a counterpart of the Thurston-Fried-McMullen "fibered face" theory in the setting of free-by-cyclic groups, that is, mapping tori groups of automorphisms of finite rank free groups. We obtain information about the BNS invariant of such groups, and construct a version of McMullen's "Teichmuller polynomial" in the free-by-cyclic context. The talk is based on joint work with Chris Leininger and Spencer Dowdall.

Decompositions of Grothendieck polynomials (Oliver Pechenik, March 8, 2017):
Finding a combinatorial rule for the Schubert structure constants in the K-theory of flag varieties is a long-standing problem. The Grothendieck polynomials of Lascoux and Sch├╝tzenberger (1982) serve as polynomial representatives for K-theoretic Schubert classes, but no positive rule for their multiplication is known outside of the Grassmannian case.
We contribute a new basis for polynomials, give a positive combinatorial formula for the expansion of Grothendieck polynomials in these "glide polynomials", and provide a positive combinatorial Littlewood-Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. A specialization of the glide basis recovers the fundamental slide polynomials of Assaf and Searles (2016), which play an analogous role with respect to the Chow ring of flag varieties. Additionally, the stable limits of another specialization of glide polynomials are Lam and Pylyavskyy's (2007) basis of multi-fundamental quasisymmetric functions, K-theoretic analogues of I. Gessel's (1984) fundamental quasisymmetric functions. Those glide polynomials that are themselves quasisymmetric are truncations of multi-fundamental quasisymmetric functions and form a basis of quasisymmetric polynomials. (Joint work with D. Searles).

The Witt group of surfaces and 3-folds (Chuck Weibel, March 1, 2017):
If V is an algebraic variety, the Witt group is formed from vector bundles equipped with a nondegenerate symmetric bilinear form. When it has dimension <4, it embeds into the more classical Witt group of the function field (Witt 1934). When V is defined over the reals, versions of the discriminant and Hasse invariant enable us to determine W(V).

Equivariant Quantum Cohomology of the Odd Symplectic Grassmannian (Ryan Shifler, February 23, 2017):
The odd symplectic Grassmannian IG:=IG(k, 2n+1) parametrizes k dimensional subspaces of $\mathbb{C}^{2n+1}$ which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on IG with two orbits, and IG is itself a smooth Schubert variety in the submaximal isotropic Grassmannian IG(k, 2n+2). We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of IG, i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case k=2, and it gives an algorithm to calculate any quantum multiplication in the equivariant quantum cohomology ring. The current work is joint with L. Mihalcea.

Fall 2016

Vector bundles of conformal blocks on the moduli space of curves (Angela Gibney, December 14, 2016):
In this talk I will introduce the moduli space of curves and a class of vector bundles on it. I'll discuss how these bundles, which have connections to algebraic geometry, representation theory, and mathematical physics, tell us about the moduli space of curves, and vice versa, focusing on just a few recent results and open problems.

Geometry and the arithmetic of algebraic structures (Daniel Krashen, December 7, 2016):
Algebraic structures, such as central simple algebras and quadratic forms, play an important role in understanding the arithmetic of fields. In this talk, I will explore the use of homogeneous varieties in working with these structures, examining in particular the splitting fields of central simple algebras, and the problem of determining the maximal dimension of anisotropic quadratic forms over a given field.

Double Gerstenhaber algebras of noncommutative poly-vector fields (Semeon Artamonov, November 30, 2016):
I will first review the algebra of poly-vector fields and differential forms in noncommutative geometry, and specific features of this generalization of conventional (commutative) differential geometry.

In the second part of my talk I will focus on noncommutative symplectic forms and noncommutative Poisson geometry. This is where the double Gerstenhaber algebra of noncommutative poly-vector fields appears. I will show that use of skew-symmetric properties allows us to substantially simply the definition.

Representations of p-DG 2-categories (Robert Laugwitz, November 16, 2016):
2-representations for k-linear 2-categories with certain finiteness conditions were studied in a series of papers by Mazorchuk-Miemietz 2010-2016. A central idea is the construction of categorifications of simple representations (so-called simple transitive 2-representations) as 2-cell representations (inspired by the Kazhdan-Lusztig cell theory to construct simple representations for Hecke algebras).
   This talk reports on joint work with V. Miemietz (UEA) adapting this 2-representation theory to a p-dg enriched setting. This approach is motivated by recent results on the categorification of small quantum groups at roots of unity (by Elias-Qi) which uses techniques from Hopfological algebra developed by Khovanov-Qi.

Rees-like Algebras and the Eisenbud-Goto Conjecture (Jason McCullough, November 9, 2016):
Regularity is a measure of the computational complexity of a homogeneous ideal in a polynomial ring. There are examples in which the regularity growth is doubly exponential in terms of the degrees of the generators but better bounds were conjectured for "nice" ideals. Together with Irena Peeva, I discovered a construction that overturns some of the conjectured bounds for "nice" ideals - including the long-standing Eisenbud-Goto conjecture. Our construction involves two new ideas that we believe will be of independent interest: Rees-like algebras and step-by-step homogenization. I'll explain the construction and some of its consequences.

The Intricate Maze of Graph Complexes (Vasily Dolgushev), November 2, 2016):
I will talk about several families of cochain complexes "assembled from" graphs. Although these complexes (and their generalizations) are easy to define, it is very hard to get information about their cohomology spaces. I will describe links between these graph complexes, finite type invariants of knots, the Grothendieck-Teichmueller Lie algebra, deformation quantization and the topology of embedding spaces. I will conclude my talk with several very intriguing open questions.

K-theoretic Schubert calculus (Oliver Pechenik, October 26, 2016):
The many forms of the celebrated Littlewood-Richardson rule give combinatorial descriptions of the product structure of Grassmannian cohomology. Anders Buch (2002) was the first to extend one of these forms to the richer world of K-theory. I will discuss joint work with Alexander Yong on lifting another form from cohomology to K-theory. This latter form has the advantage of extending further to give the first proved rule in torus-equivariant K-theory, as well as partially extending to the case of isotropic Grassmannians.

Quantum Kostka and the rank one problem for $\mathfrak{sl}_{2m}$ (Natalie Hobson, October 19, 2016):
In this talk we will define and explore an infinite family of vector bundles, known as vector bundles of conformal blocks, on the moduli space M0,n of marked curves. These bundles arise from data associated to a simple Lie algebra. We will show a correspondence (in certain cases) of the rank of these bundles with coefficients in the cohomology of the Grassmannian. This correspondence allows us to use a formula for computing "quantum Kostka" numbers and explicitly characterize families of bundles of rank one by enumerating Young tableau. We will show these results and illuminate the methods involved.

Elliptic Curves and Modular Forms (Ed Karasiewicz, October 12, 2016):
The Modularity Theorem describes a relationship between elliptic curves and modular forms. We will introduce some of the concepts needed to describe this relationship. Time permitting we will discuss some applications to certain diophantine equations.

Euler characteristics in cominuscule quantum K-theory (Sjuvon Chung, October 5, 2016):
Equivariant quantum K-theory is a common generalisation of algebraic K-theory, equivariant cohomology and quantum cohomology. We will present a brief overview of the theory before we discuss recent results on three peculiar properties of equivariant quantum K-theory for cominuscule flag varieties. This is joint work with Anders Buch.

Quaternionic Discrete Series (Zhuohui Zhang, September 28, 2016):
I will give a brief introduction to the construction and geometric background of quaternionic discrete series, and how to study them based on examples. Quaternionic discrete series are representations of a real Lie group $G$ which can be realized on a Dolbeault cohomology group of the twistor space of the symmetric space of $G$.

What is a meromorphic open string vertex algebra? (Fei Qi, September 21, 2016):
A meromorphic open string vertex algebra (MOSVA hereafter) is, roughly speaking, a noncommutative generalization of a vertex algebra. We hope that these algebras and representations will provide a starting point for a new mathematical approach to the construction of nonlinear sigma models in two dimensions.

Since this is a seminar aimed at the general audience, I'll start by explaining the notion of vertex algebra, as well as the physical meaning. Then I'll introduce the notion of a MOSVA and the physical meaning. Hopefully there will be some time to explain what I have done.

Spring 2016

Noncommutative Poisson Geometry (Semeon Artamonov, April 27, 2016):
One of the major ideas of the noncommutative geometry program consists of replacing the algebra of smooth functions on a manifold with some general associative (not necessary commutative) algebra. It appears that a lot of tools of conventional differential and algebraic geometry can be translated to the noncommutative world. In my talk I will focus on an implication of the noncommutative geometry program to the Poisson manifolds.

I will start by reviewing an $H_0$-Poisson structure --- a noncommutative analog of the Poisson bracket and related notion of double Poisson brackets. We will see how an $H_0$-Poisson structure descends to a usual Poisson bracket on the moduli space of representations of the underlying associative algebra. I will then show how one can substantially modify definition of double Poisson bracket by M. Van den Bergh to provide a number of new nontrivial examples.

Auslander-Reiten quivers of finite-dimensional algebras (Rachel Levanger, March 9, 2016):
We summarize the construction of Auslander-Reiten quivers for finite-dimensional algebras over an algebraically closed field. We give an example in the category of commutative diagrams of vector spaces.

Elliptic genera of singular varieties and related topics (Lev Borisov, March 9, 2016):
A two-variable (Krichever-Hohn) elliptic genus is an invariant of complex compact manifolds. It associates to such manifold $X$ a function in two variables. I will describe the various properties of elliptic genus. In particular, I will explain why it is a (weak) Jacobi modular form if the canonical class of $X$ is numerically trivial. I will then talk about extensions of the elliptic genus to some singular varieties.

The talk is aimed at graduate students. In particular, while some familiarity with Chern classes would be useful, I will introduce the necessary notions during the talk.

Relative Cartier divisors (Chuck Weibel, March 2, 2016):
If $B/A$ is a commutative ring extension, we consider the group $I(B/A)$ of invertible $A$-submodules of $B$. If $A$ is a domain and $B$ is its field of fractions, this is the usual Cartier divisor group. The group $I(B[x]/A[x])$ has a very interesting structure, one which is related to $K$-theory.

Arithmetic constructions of hyperbolic Kac-Moody groups (Lisa Carbone, Feb. 17, 2016):
Tits defined Kac-Moody groups over commutative rings, providing infinite dimensional analogues of the Chevalley-Demazure group schemes. Tits' presentation can be simplified considerably when the Dynkin diagram is hyperbolic and simply laced. In joint work with Daniel Allcock, we have obtained finitely many generators and defining relations for simply laced hyperbolic Kac-Moody groups over $\mathbb{Z}$. We compare this presentation with a representation theoretic construction of Kac-Moody groups over $\mathbb{Z}$. We also present some preliminary results with Frank Wagner about uniqueness of representation theoretic hyperbolic Kac-Moody groups.

Generalized adjoint actions (Volodia Retakh, Feb. 3, 2016:
We generalize the classical formula for expanding the conjugation of $y$ by $\exp(x)$ by replacing $\exp(x)$ with any formal power series. We also obtain combinatorial applications to $q$-exponentials, $q$-binomials, and Hall-Littlewood polynomials.
(This is joint work with A. Berenstein from U. of Oregon.)

Symmetrization in tropical algebra (Louis Rowen, Jan. 21, 2015):
Tropicalization involves an ordered group, usually taken to be $(\mathbb R, +)$ or $(\mathbb Q, +)$, viewed as a semifield. Although there is a rich theory arising from this viewpoint, idempotent semirings possess a restricted algebraic structure theory, and also do not reflect important valuation-theoretic properties, thereby forcing researchers to rely often on combinatoric techniques.
A max-plus algebra not only lacks negation, but it is not even additively cancellative. We introduce a general way to artificially insert negation, similar to group completion. This leads to the possibility of defining many auxiliary tropical structures, such as Lie algebras and exterior algebras, and also providing a key ingredient for a module theory that could enable one to use standard tools such as homology.

Fall 2015

Representations of finite groups and applications (Pham Huu Tiep, Dec. 7, 2015):
In the first part of the talk we will survey some recent results on representations of finite groups. In the second part we will discuss applications of these results to various problems in group theory, number theory, and algebraic geometry.

Relative Cartier divisors and polynomials (Charles Weibel, Nov. 11, 2015):
If A is a subring of a commutative ring B, a relative Cartier divisor is an invertible A-submodule of B. These divisors form a group $I(A,B)$ related to the units and Picard groups of A and B. We decompose the groups $I(A[t],B[t])$ and $I(A[t,1/t],B[t,1/t])$ and relate this construction to the global sections of an étale sheaf. This is joint work with Vivek Sadhu.

Automorphisms of extremal codes (Gabriele Nebe, Nov. 4, 2015):
Extremal codes are self-dual binary codes with largest possible minimum distance. In 1973 Neil Sloane published a short note asking whether there is an extremal code of length 72. Since then many mathematicians search for such a code, developing new tools to narrow down the structure of its automorphism group. We now know that, if such a code exists, then its automorphism group has order ≤5.

The talk will survey some aspects of this ongoing search. The methods for studying this question involve explicit and constructive applications of well known classical theorems in algebra and group theory, for instance Conway's and Pless' application of Burnside's orbit counting theorem and quadratic reciprocity dating back to the 1980's. More recent and partly computational methods are based on representation theory of finite groups.

Noncommutative Cross Ratios (Volodia Retakh, Oct. 28, 2015):
This is an introductory talk aimed at graduate students. We will introduce cross ratios and use them to define a noncommutative version of the Shear coordinates used in theoretical physics.

Quantization of Fock and Goncharov's canonical basis (Dylan Allegratti, Oct. 21, 2015):
In a famous paper from 2003, Fock and Goncharov defined a version of the space of $PGL_2(\mathbb C)$-local systems on a surface and showed that the algebra of functions on this space has a canonical basis parametrized by points of a dual moduli space. This algebra of functions can be canonically quantized, and Fock and Goncharov conjectured that their canonical basis could be deformed to a canonical set of elements of the quantized algebra. In this talk, I will describe my recent work with Hyun Kyu Kim proving Fock and Goncharov's conjecture.

Introduction to A-D-E singularities (Lev Borisov, Oct. 14, 2015):
This is an introductory talk aimed at graduate students. ADE singularities are remarkable mathematical objects which are studied from multiple perspectives. They are indexed by the so-called Dynkin diagrams $A_n$, $D_n$, $E_6$, $E_7$, $E_8$ and can be viewed as quotients of a two-dimensional complex space $\mathbb C^2$ by a finite subgroup of the special linear group $SL_2(\mathbb C)$. I will explain this correspondence as well as the relationship between ADE singularities and the Platonic solids.

Monoids, monoid rings and monoid schemes (Chuck Weibel, Oct. 7, 2015):
This is an introductory talk aimed at graduate students. If $A$ is a pointed abelian monoid, we can talk about the topological space of prime ideals in $A$, the monoid ring $k[A]$ and the topological space Spec(k[A]). Many of the theorems about commutative rings have analogues for monoids, and just as schemes are locally Spec(R), we can define monoid schemes. I will explain some of the neaterr aspects of this dictionary.

Spring 2015

The Index Map and Reciprocity Laws for Contou-Carrère Symbols
(Jesse Wolfson, Feb. 4, 2015):

In the 1960s, Atiyah and Janich constructed a natural "index" map from the space of Fredholm operators on Hilbert space to the classifying space of topological K-theory. In joint work with Oliver Braunling and Michael Groechenig, we construct an analogous index map in algebraic K-theory. The index map allows us to relate the Contou-Carrère symbol, a local analytic invariant of families of schemes, to algebraic K-theory. Using this, we prove reciprocity laws for Contou-Carrère symbols in all dimensions. This extends previous results, of Anderson and Pablos Romo in dimension 1, and of Osipov and Zhu, in dimension 2.

Zero divisors in the Grothendieck ring of varieties (Lev Borisov, Feb. 25, 2015):
I will explain the motivation and results of my recent preprint that shows that the class of the affine line is a zero divisor in the Grothendieck ring of varieties.

Fall 2014

Semiclassical approximation to noncommutative Riemannian geometry
(Edwin Beggs, Sept. 17, 2014):

I will consider the first order deformation of a Riemannian manifold, including the vector bundles, differential calculus and metric. One example will be the Schwarzschild solution, which illustrates that not all the properties of the classical case can be simply carried into the quantum case. The other example is quantising the Kahler manifold, complex projective space. This case is much simpler, and here the complex geometry is also preserved. I will end with some comments on the connection between noncommutative complex geometry and noncommutative algebraic geometry.

Equivariant quantum cohomology and puzzles (Anders Buch, Sept. 24, 2014):
The "classical equals quantum" theorem states that any equivariant Gromov-Witten invariant (3 point, genus zero) of a Grassmann variety can be expressed as a triple intersection of Schubert classes on a two-step partial flag variety. An equivariant triple intersection on a two-step flag variety can in turn be expressed as a sum over puzzles that generalizes both Knutson and Tao's puzzle rule for Grassmannians and the cohomological puzzle rule for two-step flag varieties. These results together give a Littlewood-Richardson rule for the equivariant quantum cohomology of Grassmannians. I will speak about geometric and combinatorial aspects of this story, which is based on papers with Kresch, Purbhoo, Mihalcea, and Tamvakis.

Cut and paste approaches to rationality of cubic fourfolds (Lev Borisov, Oct. 8, 2014):
A random 4-dimensional hypersurface of degree 3 is widely expected to be nonrational, but no proof of this statement currently exists. Moreover, there is no clear understanding precisely which such fourfolds are ratonal. An approach to this problem developed in a recent preprint of Galkin and Shinder gives an unexpected necessary condition of rationality modulo a variant of cancellation conjecture. This is a surprisingly short and clean argument that involves the Grothendieck ring of varieties. I will aim to make my talk accessible to the audience with very limited algebraic geometry background.

The Witt group of real varieties (Chuck Weibel, Oct. 15, 2014):
We approximate the Witt groups of a variety V over the reals, using a topological invariant: the Witt groups of Real vector bundles on the space of complex points of V. This is a better approximation than one might expect, and has the advantage of being finitely generated. This is joint work with Max Karoubi.

Jacobians of modular curves (Ed Karasiewicz, Oct. 22, 2014):
We study the Jacobian variety of a modular curve $C$ over an elliptic curve, and its Hecke operators. The goal is to show that the $L$-function of a weight 2 cusp form for $C$ is the same as the $L$-function of the elliptic curve.

A Modular Operad of Embedded Curves (Charles Siegel, Oct. 29, 2014):
Modular operads were introduced by Getzler and Kapranov to formalize the structure of gluing maps between moduli of stable marked curves. We present a construction of analogous gluing maps between moduli of pluri-log-canonically embedded marked curves, which fit together to give a modular operad of embedded curves. This is joint work with Satoshi Kondo and Jesse Wolfson.

Some non-compact Riemann surfaces branched over three points
(Marvin Tretkoff, Nov. 12, 2014):

Recall that the Riemann surface of the multi-valued function $\log(z)$ is an infinite sheeted covering of the z-sphere branched over the two points $z=0$ and $z=\infty$ and has an ''infinite spiral ramp'' over each of them. Consequently, its monodromy group is infinite cyclic.
Today, we construct Riemann surfaces as infinite sheeted coverings of the z-sphere that are branched over precisely three points on the z-sphere. Moreover, each of these Riemann surfaces has a single ''infinite spiral ramp'' over each of its branch points. The monodromy groups of such surfaces are infinite two generator groups of permutations of the set of integers. Our construction yields many non-isomorphic groups with varying algebraic properties. In this lecture, we shall discuss one of these in some detail.

Property T for Kac-Moody groups (Ashley Rall, Nov. 19, 2014):
I will give a brief introduction to Kac-Moody groups, infinite dimensional analogues of Chevalley groups, and Kazhdan's property (T) and then discuss joint work with Mikhail Ershov establishing property (T) for Kac-Moody groups over rings. We expand upon previous results by Dymara and Januszkiewicz establishing property (T) for Kac-Moody groups over finite fields and by Ershov, Jaikin, & Kassabov establishing property (T) for Chevalley groups over commutative rings to prove that given any indecomposable 2-spherical generalized Cartan matrix A there is an integer m (depending solely on A) such that if R is a finitely generated commutative unital ring with no ideals of index less than m then the Kac-Moody group over R associated to A property (T).

Sieve methods in group theory (Alex Lubotsky, Dec. 3, 2014):
The sieve methods are classical methods in number theory. Inspired by the 'affine sieve method' developed by Sarnak, Bourgain, Gamburd and others, as well as by works of Rivin and Kowalsky, we develop in a systemtic way a 'sieve method' for group theory. This method is especially useful for groups with 'property tau'. Hence the recent results of Breuillard-Green-Tao, Pyber-Szabo, Varju and Salehi-Golsefidy are very useful and enables one to apply them for linear groups.
We will present the method and some of its applications to linear groups and to the mapping class groups. [Based on joint with Chen Meiri (JAMS) and with Lior Rosenzweig (to appear in Amer. J. of Math.) ].

Spring 2014 Seminars (Wednesdays at 2:00 in H525)
Abstracts and Details for Spring 2014 seminars are located at

Fall 2013

Torsion Theory and Slice Filtration of Homotopy Invariant Sheaves With Transfers
(Knight Fu, Sept. 16, 2013):

Torsion Theory makes important contributions to the study of modules over a ring. It also plays an important role in constructing the quotient of an abelian category by a "torsion" subcategory.
Applying torsion theory to the category of homotopy invariant sheaves with transfers, we show how a sequence of co-radical functors gives rise to two filtrations --- one increasing, one decreasing --- of the category. We conjecture that the obtained structure ought to be the "slice filtration" on the category, and show how the filtrations are compatible with the slice filtration on Voevodsky's derived category of motives.

Spring 2013

Equations, syzygies, and vector bundles (Daniel Erman, Jan. 24, 2013):
For a system of polynomial equations, it has long been known that the relations (or syzygies) among the polynomials provide geometric information about the corresponding projective variety. I will describe a collection of new ideas about how to study syzygies, and how these lead to classification results and a duality between syzygies and vector bundles.

Equivariant Schubert calculus: positivity, formulas, applications
(David Anderson, Jan. 30, 2013):

Schubert's enumerative calculus is the subject of Hilbert's 15th problem. It is a technique for solving problems of enumerative geometry; for example, how many conics are tangent to five given conics? In its modern formulation, Schubert calculus concerns computations in the cohomology rings of Grassmannians, flag varieties, and related spaces. These spaces carry large group actions, which can be used to both refine and simplify the computations. The cohomology calculations can be modeled by multiplication of polynomials, and a central role is played by these polynomial representatives. Formulas for these polynomials are of both theoretical and computational interest. In this talk, I will survey recent developments in this subject, including some new formulas and applications.

What is a Derivator? (Chuck Weibel, Feb. 6, 2013):
As the name implies, this is an introductory talk. Derivators were introduced in 1983 by Grothendieck in a 600-page manuscript, and refined in his 2000-page manuscript in 1991. They are designed to enhance triangulated categories, and have recently been used in the study of non-commutative algebraic geometry.

A geometric approach to noncommutative Laurent phenomenon (V. Retakh, Feb. 13, 2013):
A composition of birational maps given by Laurent polynomials need not be a Laurent polynomial. When it does, we talk about the Laurent phenomenon. A large variety of examples of the Laurent phenomena for commuting variables comes from the theory of cluster algebras. Much less is known in the noncommutative case. I will present a number of the noncommutative Laurent phenomenoma of a "geometric origin." This is a joint work with A. Berenstein.

Dynamics and surjectivity of some word maps on SL(2,q)
Tatiana Bandman, Feb. 20, 2013):

I will speak about a geometric approach, based on the classical trace map, for investigating dynamics, surjectivity and equidistribution of word maps on groups PSL(2,q) and SL(2,q). It was also used for a characterization of finite solvable groups by two-variable identities.

Strongly Dense Subgroups of Algebraic Groups (Robert Guralnick, Feb. 27, 2013):
Let G be a simple algebraic group. A free finitely generated subgroup H of G is called strongly dense in G if every nonabelian subgroup of H is Zariski dense in G. We will discuss joint work with Breuillard, Green and Tao which shows that strongly dense subgroups exist (over sufficiently large fields) and some recent improvements on this by Brueillard, Guralnick and Larsen. This has applications to finding Cayley graphs of finite simple groups of Lie type and some results on generation of finite simple groups of Lie type. Using these ideas, we can also improve on results of Borel and Deligne-Sullivan related to the Hausdorff-Banach-Tarski paradox.

The 3 main problems in the braid group (Mina Teicher, March 13, 2013):
These are: The Word Problem, The Conjugacy Problem and the Hurwitz Equivalence Problem. I shall present the questions, some answers and, time permitted, also an application to Cryptography.

Intersection theory on singular varieties (Joe Ross, April 3, 2013):
Whereas algebraic cycle classes may be multiplied on a smooth variety, this is not in general possible on a singular variety. In topology, the intersection homology groups of Goresky-MacPherson provide interesting invariants of singular spaces. Intersection homology sits in between singular cohomology and homology, and admits natural pairings generalizing the product structure on the singular homology of a smooth manifold.
I will propose an algebraic analogue of intersection homology which sits in between the algebraic cocycles of Friedlander-Lawson and the classical Chow groups. In some special cases these "intersection Chow groups" admit pairings. This is joint work with Eric Friedlander.

On Hilbert modular threefolds of discriminant 49 (Lev Borisov, April 10, 2013):
I will talk about joint work with Paul Gunnells in which we explicitly describe a certain Hilbert modular threefold as a hypersurface in a weighted projective space. As a byproduct we find an octic surface in P^3 with 84 singularities of type A_2.

Implicit systems of differential equations (Freya Pritchard, April 24, 2013)
We will consider implicit systems that are given by polynomial relations on the coordinates of the indeterminate function and the coordinates of the time derivative of the indeterminate function. For such implicit system of differential-algebraic equations, we will be concerned with algebraic constraints such that on the algebraic variety determined by the constraint equations the original implicit system of differential equations has an explicit representation.
Our approach to such systems is algebraic. Although there have been a number of articles that approach implicit differential equations algebraically, all such approaches have relied heavily on linear algebra. The approach that we will consider is different in that we have no linearity requirements at all, instead we shall rely on classical algebraic geometry. In particular we will use birational mappings to produce an explicit system of differential equations and an algebraic variety of possible initial values.

Structure of Chevalley groups over rings (Alexei Stepanov, May 1, 2013)
Let G be a Chevalley group scheme with elementary group E. Using a localization procedure to reduce to the well understood case of local rings, we study the following problems over a commutative ring R:
a) Normality of E(R) and commutator formulas;
b) Nilpotent structure of K1=G(R)/E(R)
c) bounded word length in E(R); and
d) normal subgroups of G(R).

Fall 2012

Curve neighborhoods Anders Buch, Oct. 3, 2012):
Given a generalized flag manifold X = G/P, a Schubert variety X(w), and a degree d, consider the set of points that can be reached from X(w) by a rational curve of degree d, i.e. the union of all rational degree d curves through X(w). It turns out that the Zariski closure of this set is a larger Schubert variety, which is important for many aspects of the quantum cohomology of X, including the quantum Chevalley formula and the smallest q-degree in the quantum product of two Schubert classes. I will give a very explicit description of this "curve neighborhood" of the Schubert variety in terms of the Hecke product of Weyl group elements, and use it to give a simple proof of the (equivariant) quantum Chevalley formula. This is joint work with Leonardo Mihalcea.

Computations in intersection theory (Dan Grayson, Oct. 17, 2012):
This is joint work with Alexandra Seceleanu and Michael E. Stillman. We describe Groebner bases for the ideals of relations between the Chern classes of the tautological bundles on partial flag bundles, and show how the result can be used to enable practical computation of intersection numbers in the "Macaulay2" package "Schubert2". We also generalize the result to cover isotropic flag bundles.

On the slice filtration for hermitian K-theory (Oliver Rondigs, November 7, 2012):
Let F be a perfect field of characteristic different from 2. In joint work with Paul Arne Ostvaer, we describe the slices of hermitian K-theory and higher Witt-theory in the motivic stable homotopy category of F. Applications include computations of hermitian K-groups and Witt groups for number fields and projective spaces, as well as a different perspective on Milnor's conjecture on quadratic forms.

Injectivity property of etale H^1, non-stable K_1, and other functors (Anastasia Stavrova, December 5, 2012):
The first term of Gersten conjecture for K-theory claims the injectivity of the map K_i(R)→K_i(K) for any regular semilocal ring R with field of fractions K. The same statement with K_i replaced by the etale cohomology functor H^1(-,G), where G a reductive algebraic group, is known as the Grothendieck-Serre conjecture. The latter conjecture was recently settled by I. Panin et al. under the assumption that R contains an infinite perfect field. We discuss how essentially the same argument carries over to non-stable K_1 and similar functors.

Spring 2012

Exhausting quantization procedures (Vasily Dolgashev, Jan. 25, 2012):
Deformation quantization is a procedure which assigns a formal deformation of the associative algebra of functions on a variety to a Poisson structure on this variety. Such a procedure can be obtained from Kontsevich's formality quasi-isomorphism and, it is known that, there are many homotopy inequivalent formality quasi-isomorphisms. I propose a framework in which all homotopy classes of formality quasi-isomorphisms can be described. More precisely, I will show that homotopy classes of "stable" formality quasi-isomorphisms form a torsor for the group exp(H°(GC)), where GC denotes the full graph complex. The group exp(H°(GC)) is isomorphic to the Grothendieck-Teichmueller group which is, in turn, related to moduli of curves and to theory of motives.

Shift Equivalence and Z[t]-modules (Chuck Weibel, February 8, 2012):
Shift equivalence is an equivalence relation on nxn matrices (say over Z). Such a matrix T may be regarded as defining a Z[t]-module structure on a free abelian group, and shift equivalence translates into the assertion that the modules become isomorphic over Z[t,1/t]. This talk is a description of a weaker equivalence relation related to class groups of number fields.

K-theory of minuscule varieties (Anders Buch, February 22, 2012):
Thomas and Yong have conjectured a Littlewood-Richardson rule for the K-theory of any minuscule homogeneous space, based on counting tableaux that rectify to a certain superstandard tableau. This conjecture has been proved for Grassmannians of type A and maximal orthogonal Grassmannians, but it fails for the Freudenthal variety of type E7. I will speak about a fix that replaces the superstandard tableaux with minimal increasing tableaux. These tableaux have several other combinatorial advantages, for example they make it possible to recognize which tableaux should be counted without rectifying them. This is joint work with Matthew Samuel.

From algebra to category theory: a first approach to fusion categories
(Julia Plavnik, February 29, 2012):

A good way of thinking about category theory is that it is a refinement (or "categorification") of ordinary algebra. In other words, there is a dictionary between these two subjects, such that usual algebraic structures are recovered from the corresponding categorical structures by passing to the set of isomorphism classes of objects (Etingof, Gelaki, Nikshych and Ostrik).

The idea of this talk is to introduce and motivate the notion of fusion category. We shall give some basic definitions and examples that help us understand this structure. We shall introduce the ideas of gradings, solvability and nilpotency for fusion categories and we shall connect it to the corresponding ideas for groups. We shall also discuss some results concerning to the structure of fusion categories with restrictions on the Frobenius-Perron dimensions of its simple objects.

Invariants of Matrix Factorizations (Mark Walker, March 21, 2012):
Given, for example, a polynomial $f(x_1,...,x_n)$ with complex coefficients, a matrix factorization of f is a pair of r x r matrices of polynomials (A, B) satisfying $AB = f I_r = BA$. Introduced over 30 years ago by Eisenbud in the context of studying projective resolutions of modules over hyper-surfaces, there has been a revival of interest in matrix factorizations lately, as connections with mathematical physics and knot theory have emerged. I will discuss some recent progress in understanding certain fundamental invariants of matrix factorizations.

Combinatorial aspects of toric mirror symmetry (Lev Borisov, March 28, 2012):
I will review the combinatorial aspects of toric mirror symmetry. In particular, I will focus on the new phenomena one encounters when dealing with complete intersections as opposed to hypersurfaces.

Noncommutative Laurent Phenomena (Vladimir Retakh, April 11, 21012:
I will discuss the Laurent phenomenon for noncommuting variables. A good example is the cluster conjecture of Kontsevich. I will present a proof of the conjecture, recently obtained by A. Berenstein and me.

Symmetric subgroup orbit closures on flag varieties as universal degeneracy loci
(Ben Wyser, April 18, 2012:

Suppose that G is one of the classical groups SL(n,C), SO(n,C) or Sp(2n,C), and that K is a symmetric subgroup of G --- that is, the fixed points of an involution of G. The group K has finitely many orbits on the flag variety G/B, and the geometry of these orbits and their closures is closely connected to the theory of Harish-Chandra modules for a certain real form of G. Their representation-theoretic interest aside, such orbit closures are, in a sense, generalizations of Schubert varieties, and most questions one has about Schubert varieties can equally well be posed about these more general objects.

We will describe a method for computing formulas for the S-equivariant fundamental classes of such orbit closures, where S is a maximal torus of K. The main idea is to use equivariant localization and the self-intersection formula to "guess" formulas for the classes of closed orbits, and then to compute formulas for the remaining orbit closures using divided difference operators. In type A, we will also describe how these formulas can be interpreted as Chern class formulas for classes of certain types of degeneracy loci involving a vector bundle over a scheme which is equipped with a complete flag of subbundles and an additional structure determined by K. This is analogous to (and motivated by) work of W. Fulton on Schubert varieties in flag bundles, their role as universal degeneracy loci for maps of flagged vector bundles, and connections between that work and the torus-equivariant cohomology of the flag variety described by W. Graham.

Fall 2011

The Schottky Problem and genus 5 curves (Charles Siegel, Sept. 14, 2011):
The relationship between algebraic curves and abelian varieties has a long and classical history. One of the most fundamental open problems is determining when an abelian variety is the Jacobian of some curve. We will discuss some of the history of the problem, as well as new results in the case of genus 5 curves.

Congruence subgrous of lattices in rank 2 Kac-Moody groups over finite fields
(Abid Ali, Sept. 28, 2011):

Let G be a complete Kac-Moody group of rank 2 over a finite field, and let B— denote the non-uniform lattice subgroup generated by the "diagonal subgroup" and all negative real root groups. We define and construct congruence subgroups of B—. This is joint work with Lisa Carbone.

Cluster algebras and combinatorics of rigid objects in 2 Calabi-Yau categories
(Raika Dehy, October 4 and 18, 2011):

This talk is motivated by the representation-theoretic approach to Fomin-Zelevinsky's cluster algebras. In this approach a central role is played by certain 2-Calabi-Yau categories and by combinatorial invariants associated with their rigid objects (objects with no self-extensions).

I shall recall the definition of cluster algebras and how to construct the cluster categories associated with them (the latter are 2-Calabi-Yau categories). Then I will introduce the combinatorial invariant that will help prove part of the conjectures on g-vectors associated to cluster variables.

Giambelli formulas for orthogonal Grassmannians (Anders Buch, October 26, 2011):
Let X be an orthogonal Grassmannian, defined as the set of all isotropic subspaces of a given dimension in a complex vector space equipped with an orthogonal bilinear form. The cohomology ring H^*(X) has a basis consisting of Schubert classes; products of these classes have applications in enumerative geometry and are the main objects of study in Schubert calculus. The cohomology ring H^*(X) can also be understood in terms of generators and relations, where the generators are certain special Schubert classes. A Giambelli formula means an expression of an arbitrary Schubert class as a polynomial in special Schubert classes. The Schubert classes of an ordinary Grassmann variety can be expressed as determinants of matrices of special Schubert classes, and the Schubert classes of a maximal orthogonal Grassmannian can be written as Pfaffians. I will speak about new Giambelli formulas for submaximal orthogonal Grassmannians that is expressed in terms of raising operators and interpolate between the above cases. This is joint work with A. Kresch and H. Tamvakis.

On Fourier-Mukai type functors (Alice Rizzardo, November 2, 2011):
Orlov showed in 1997 that all exact, fully faithful functors between the bounded derived categories of two smooth projective varieties are isomorphic to a Fourier-Mukai transform. In this talk we will discuss a class of functors that are not full or faithful and still satisfy the above result.

Comparison of Dualizing Complexes (Changlong Zhong, November 9, 2011):
In this talk I will introduce four dualizing complexes defined by M. Spiess, T. Moser, S. Bloch (duality proved by T. Geisser) and K. Sato, and compare them in the derived category. We show that Bloch's complex is quasi-isomorphic with all three, in the situation when they are properly defined (and assuming some well-known conjectures).

Homotopical Methods in Algebraic Geometry (Pablo Pelaez, November 30, 2011):
Algebraic Topology and Algebraic Geometry throughout the years have shared common methods and enriched each other. The work of Morel-Voevodsky gives a natural categorical framework to import some standard methods from homotopy theory into algebraic geometry. The aim of this talk is to describe some concrete examples.

Spring 2011

Syzygies of binomial ideals and toric Eisenbud-Goto conjecture (Lev Borisov, April 6, 2011):
Let p1,...,pk be a collection of points with integer coordinates. Denote their convex hull by Δ. For every integer n consider the subset inside the multiple nΔ which consists of lattice points that can be written as sums of n of the pi. Typically, some lattice points of nΔ will be missing from this set. The toric Eisenbud-Goto conjecture gives a certain measure of control over the sets of missing points. I will give an elementary introduction to the conjecture, which is still open for the case of six points on the plane.

Cyclic homology, simplicial rapid decay algebras, and applications to K*t(l¹(G))
(Crichton Ogle, April 13, 2011):

Using techniques developed for studying polynomially bounded cohomology, we show that the assembly map for K*t(l¹(G)) is rationally injective for all finitely presented discrete groups G. This verifies the l¹-analogue of the Strong Novikov Conjecture for such groups. The same methods show that the Strong Novikov Conjecture for all finitely presented groups can be reduced to proving a certain (conjectural) rigidity of the topological cyclic chain complex CC*t(HCM(F)) where F is a finitely-generated free group and HCM(F) is the "maximal" Connes-Moscovici algebra associated to F.

Rational points, zero cycles of degree one, and A^1-homotopy theory (Christian Haesemeyer, Feb. 16, 2011):
A system of polynomial equations over a field F may have solutions in a collection of finite field extensions of relatively prime degree, but not have any solution in F. We will describe some examples and results known about this phenomenon, and then talk about what A^1-homotopy theory might contribute to understanding it.

Fall 2010

Quantum differential operators (Uma Iyer, Sept. 20, 2010):
In the late 1990s, Lunts and Rosenberg gave a definition of quantum differential operators on graded algebras which allow us to view the action of quantum groups on graded algebras as quantum differential operators. We present the algebras of quantum differential operators on certain graded algebras.

Monoids and algebraic geometry (Chuck Weibel, Sept. 27, 2010):
The prime ideals in an abelian monoid are the points of a topological space, and we can glue them together to get finite models of toric varieties and other objects in algebraic geometry. In this introductory talk, we give basic properties of these monoidal schemes, such as normalization, proper maps and resolution of singularities.

Dimensions of fixed point spaces of elements in linear groups (Bob Guralnick, Oct. 4, 2010)
Sizes of fixed point spaces for elements in linear groups have a long history. These types of results include the classification of psuedoreflection groups and Frobenius complements. I will talk about recent joint works with Maroti and Malle which answer two conjectures from Peter Neumann's 1966 thesis. The first conjecture has to do with the average dimension of a fixed space of an element in a finite (irreducible) linear group. The second has to do with the minimal dimension of a fixed space of some element in an irreducible (possibly infinite) linear group.

Etale cohomology operations (Chuck Weibel, Oct. 25, 2010)
In topology, the Steenrod algebra describes all natural transformations between cohomology groups H*(-,Z/p). Modifying this construction, Epstein constructed certain operations Pi on etale cohomology with coefficients Z/p, used by Raynaud in her construction of universal projective modules. We modify his construction to allow twisted coefficients such as μp, and give a complete list of all etale cohomology operations in this context.
This is joint work with Bert Guillou.

Pieri rules for the K-theory of cominuscule Grassmannians (Anders Buch, Nov. 8, 2010)
The K-theoretic Schubert structure constants of a homogeneous space G/P are known to have signs that alternate with codimension by a result of Brion. For Grassmannians of type A, these constants are computed by a generalization of the classical Littlewood-Richardson rule that counts set-valued tableaux. The K-theory ring of any Grassmann variety is generated by special Schubert classes that correspond to partitions with a single row.

I will present positive combinatorial formulas for the structure constants in products involving special Schubert classes on any cominuscule Grassmannian. Together with a result of Clifford, Thomas, and Yong, this proves a K-theoretic Littlewood-Richardson rule for maximal orthogonal Grassmannians. This is joint work with Vijay Ravikumar.

The Lie product in the continuous Lie dual of the Witt Algebra (Earl Taft, Nov. 29, 2010):
Let k be a field of characteristic zero. The simple Lie algebra W1=Der k[x], the one-sided Witt algebra, has a basis ei=x(i+1)d/dx for i at least -1. The wedges of e0 and ei satisfy the classical Yang-Baxter equation, giving W1 the structure of a coboundary triangular Lie bialgebra. Its continuous Lie dual is also a Lie bialgebra, and has been identied with the space of k-linearly recursive sequences by W. Nichols. Let f=(fn) and g=(gn) be linearly recursive sequences in the continuous linear dual. For each n, the n-th coordinate of [f,g] has been described in terms of the coordinates of f and of g, but it was an open problem to give a recursive relation satised by [f,g] in terms of recursive relations satisfied by f and by g. We give such a relation here. Analogous`results hold for the two-sided Witt algebra Der k[x,x-1].
This is joint work with Zhifeng Hao.

Motivic cohomology operations (Chuck Weibel, Dec. 6, 2010)
Voevodsky gave a description of all stable operations in motivic cohomology in 2009. However, many other unstable operations have come to light in the last decade. We determine the unstable operations. This is joint work with Bert Guillou.

Algebraic Structures from Operads (Ralph Kaufmann, Dec. 13, 2010):
There are certain classic algebraic structures like Gerstenhaber's bracket which have their origin in operads. We discuss several generalizations of these structures - notably Lie brackets, BV operators and master equations. We show how these appear naturally in operadic settings. Our general theory gives a unified framework for a diverse set of geometric and algebraic examples.

Spring 2010

Periodicity of hermitian K-groups (Max Karoubi, Feb. 1, 2010):
This is joint work with Jon Berrick and Paul Arne Ostvaer.
It has been known for a few years, essentially by the work of Voevodsky and Rost, that the algebraic K-theory of a commutative ring A with finite coefficients is periodic above the etale cohomological dimension of A. In this lecture, we show that such a ring A has also a periodic hermitian K-theory in the same range.
This essentially means that theorems about the general (infinite) linear group, such as the one proved by Rost and Voevodsky, imply similar ones for the orthogonal and symplectic groups.

Applications of stable bundles to Witt groups and Brauer groups (Ray Hoobler, Mar. 1, 2010):
Basic properties of stable bundles on a projective, smooth variety X will be outlined. These properties make maps between stable bundles quite rigid so that such bundles behave almost like elements in a basis of a vector space over a field. The picture is particularly clear for projective, smooth varieties over a finite field. This will be applied to determine generators of the Witt group of X and to show that the stable Brauer group of X is the same as the Brauer group of X.
   Familiarity with the definitions of Witt groups and Brauer groups of fields will be helpful but not essential.

Drinfeld twists and finite groups (Christian Kassel, Mar. 8, 2010):
Drinfeld twists were introduced by Drinfeld in his work on quasi-Hopf algebras. In joint work with Pierre Guillot (arXiv:0903.2807, published in IRMN), after observing that the invariant Drinfeld twists on a Hopf algebra form a group, we determine this group when the Hopf algebra is the algebra of a finite group G. The answer involves the group of class-preserving outer automorphisms of G as well as all abelian normal subgroups of G of central type.

Hopf algebras and recursive sequences (Earl Taft, Mar. 22, 2010):
Linearly recursive sequences have a bialgebra structure. Polynomially recursive(or D-finite) sequences have a topological bialgebra structure. If such a sequence is of a combinatorial nature, a formula for its coproduct can often be interpreted as a combinatorial identity. We illustrate this for the sequences whose n-th term is ((ni)(n!)) for a fixed non-negative i, where (ni) is the binomial coefficient. The resulting combinatorial identity is of an iterated Vandermonde type.

Tilting 1 (Chuck Weibel, March 29, 2010):
This is an overview of the notion of tilting, from Gelfand-Ponomarev to the 1990s. Given a ring A, an A-module T is tilting if it has finite projective dimension, Exti(T,T)=0 for i>0, and there is a resolution 0 → A → T0 → ... → Tn →0      with the Ti summands of sums of copies of T.
Then Hom(T,-) and T⊗B- determine an equivalence between Db(A) and Db(B), where B=End(T). In the associated torsion theory, the torsion modules are the quotients of direct sums of copies of T.

Tilting 2 (Chuck Weibel, April 19, 2010):
Tilting modules were introduced in the 1970s (originally by Gelfand-Ponomarev to construct reflection functors), with the restriction that the modules have projective dimension 1. We will present the main results of tilting in this setting, including the torsion theory associated with these modules.

Tilting 3 (Robert Wilson, April 26, 2010):
The modern notion of a tilting module was presented in the seminar talk Tilting 1, and is due to Cline-Parshall-Scott. Any left derived functor inducing an equivalence between A-modules and B-modules arises from a tilting module, as RHom_A(T,-).

Module Structures on the Ring of Hurwitz Series Bill Keigher, May 3, 2010):
Let k be a field of characteristic p>0. We consider monic linear homogeneous differential equations (LHDE) over the ring of Hurwitz series Hk of k. We obtain explicit recursive expressions for solutions of such equations and show that Hk admits a full a set of solutions as well. We then consider the notion of intertwining of Hurwitz series to reduce the study of solutions of an nth order equation to a system of n first order equations in a particularly simple form. For every LDHE over Hk we will associate a module (over a suitable quasifield extension of k), which is closed under the shift derivation of Hk and discuss the structure of the group of module automorphisms that commutes with the shift derivation.

Fall 2009

Differential Central Simple Algebras and Picard-Vessiot representations (Lourdes Juan, Oct. 5, 2009):
A differential field is a field K with a derivation, that is, an additive map D:K → K satisfying D(fg)=D(f)g+fD(g) for f,g in K. The field of constants C of K are the zeros of D. A differential central simple algebra (DCSA) over K is a pair (A,\mathcal D) where A is a central simple algebra and $\mathcal D$ is a derivation of A extending the derivation D of its center. Any DCSA, and in particular a matrix differential algebra over K, can be trivialized by a Picard-Vessiot (differential Galois) extension E of K. In the matrix algebra case, there is a correspondence between K-algebras trivialized by E and representations of the differential Galois group of E over K in PGLn(C) that can be interpreted as cocycles equivalent up to coboundaries. I will start with a brief introduction to differential Galois theory.

Derangements in Finite and Algebraic Groups (Bob Guralnick, Oct. 12, 2009):
A permutation on a set is called a derangement if it has no fixed points. The study of the proportion of derangements in finite transitive groups has a long history and the problem has many applications. We will discuss this as well as the analogous problem for algebraic and show the connection between the two. In particular, we will discuss recent results (joint with Fulman) about conjugacy classes in finite Chevalley groups and the solution of a conjecture made independently by Aner Shalev and Nigel Boston.

Mathematics arising from a new look at the Dedekind-Frobenius group matrix and group determinant (Ken Johnson, Oct. 19, 2009):
Frobenius invented group character theory in order to solve the problem of the factorization of the group determinant. His papers are hard to understand and when the modern methods for group representation theory were introduced his initial work was largely forgotten. To each representation of a (finite) group there is associated a polynomial which is a factor of the group determinant, and Frobenius introduced "k-characters" to describe this polynomial. Professor Gelfand has commented that perhaps physicists might benefit from looking at these polynomials. Among other places these k-characters have occurred in work of Buchstaber and Rees and also are related to work of Wiles and Taylor on "pseudocharacters" of finite dimensional representations of infinite groups.
I will describe the early work from an elementary point of view and give an account of some of the new ideas coming from it, and also indicate some of the connections with probablity.

Induced definable structure on cyclic subgroups of the free group
(Chloe Perin, Nov. 2, 2009):

Let C be a cyclic subgroup of a finitely generated free group F. We show that the intersection of a definable set D in F^n with C^n is in the Boolean algebra of cosets of subgroups of C^n. In other words, the definable structure induced by the embedding of C in F is no richer than the definable structure on C. We make extensive use of Sela's geometric techniques for studying the first-order theory of the free group, in particular of his construction of "formal solutions" to an equation.

The classfication problem for finite rank dimension groups (Paul Ellis, Nov. 9, 2009):
An unperforated partially ordered abelian group A is a dimension group if A satises the Riesz interpolation property (given a,a' ≤b,b' there is a c with a,a' ≤ c ≤b,b'). These are related to "Bratteli diagrams". Paul will discuss the difficulty of classifying them when the rank is at least 3, and show that the problem for a given rank cannot be reduced to the classification problem for a smaller rank.

Picard-Vessiot Theory (Ravi Srinivasan, Nov.16, 2009):
Let F be a characteristic zero differential field with an algebraically closed field of constants C. I will describe the construction of a Picard-Vessiot Extension (PVE) for a linear homogeneous differential equation over F. The group of differential automorphisms of a PVE fixing F is called the differential Galois group; there is a Galois correspondence between its algebraic subgroups and intermediate differential subfields. Examples of PVEs for F=C(x) with the usual derivation will be discussed, and we will also compute the differential Galois group for our examples.

Spring 2009

Stability conditions for triangulated categories (Chuck Weibel, Feb. 2, 2009):
This is an introductory survey talk.
There is a complex topological manifold, called the Stability Space, associated to any triangulated category D. It was conceived by Mike Douglass as an aspect of string theory, and made mathematical by Tom Bridgeland. Subspaces correspond to t-structures, and the stability space of the projective line is the affine complex plane.

Partitioned Differential Quasifields (Leon Pritchard, Feb. 23, 2009):
A differential quasifield is a natural generalization of a differential field in characteristic p>0. Elementary properties of differential quasifields are considered, and a generalized version of the theorem on the connection between linear independence over constants and the Wronskian is presented.

Stability conditions in Physics (Jan Manschot, March 2, 2009):
In a recent seminar (2/2/09), C. Weibel discussed recent developments on stability in (triangulated) categories. These developments are inspired by physics, in particular string theory. This introductory talk will explain the notion of stability in string theory, and how it is connected to stability in mathematics.

The Nekrasov Conjecture for Toric Surfaces (Elizabeth Gasparim, March 30, 2009):
The Nekrasov conjecture predicts a relation between the partition function for N=2 supersymmetric Yang-Mills theory and the Seiberg-Witten prepotential. For instantons on ℝ4, the conjecture was proved, independently and using different methods, by Nekrasov-Okounkov, Nakajima-Yoshioka, and Braverman-Etingof. We prove a generalized version of the conjecture for instantons on noncompact toric surfaces.

Differential Quasifields (Bill Keigher, April 13, 2009):
In a recent seminar (2/23), Leon Pritchard talked about partitioned differential quasifields.

Morphisms of cohomological field theories and behavior of Gromov-Witten invariants under quotients
(Chris Woodward, April 20, 2009):

I will talk about a "quantum non-abelian localization" conjecture that relates Gromov-Witten invariants of GIT quotients with equivariant Gromov-Witten invariants of the total space. Some special cases are proved. A key notion in the conjecture is the notion of morphism of cohomological field theories, which "complexifies" the notion of A-infinity morphism.

higher order Hochschild (co)homology (Gregory Ginot, April 27, 2009):
We will explain how one can define Hochschild (co)chain complex associated in a functorial way to any space X, CDG algebra A and A-module M. We will give several examples and applications to Adams operations and (if time permits) Brane topology.

Fall 2008

Morita Equivalence Revisited (Paul Baum, Sept. 5, 2008):
Notation: k denotes a unital algebra over the complex numbers which is commutative, finitely generated, and nilpotent-free, i.e., k is the coordinate algebra of a complex affine variety. A k-algebra is an algebra A over the complex numbers which is a k-module such that the algebra structure and the k-module structure are compatible in the evident way. Note that A is not required to be commutative. Prim(A) denotes the set of primitive ideals in A. Prim(A) is topologized by the Jacobson topology.

This talk studies an equivalence relation between k-algebras which is a weakening of Morita equivalence. If A and B are equivalent in the new equivalence relation, then A and B have isomorphic periodic cyclic homology, and Prim(A) is in bijection with Prim(B). However, the bijection between Prim(A) and Prim(B) might not be a homeomorphism. Thus the new equivalence relation permits a tearing apart of strata in the primitive ideal spaces which is not allowed by Morita equvalence. An application to the representation theory of p-adic groups will be briefly indicated. This talk is intended for non-specialists. All the basic definitions will be carefully stated.
The above is joint work with A.M.Aubert and R.J.Plymen.

Formality theorems for Hochschild (co)chains and their applications (Vasily Dolgushev, Sept. 18, 2008):
I will start my talk with a review of the algebraic operations on the pair Hochschild cochain complex and Hochschild chain complex of an associative algebra. Then I will speak about the formality theorems for these complexes. Finally I will discuss applications of these formality theorems to deformation quantization, computation of Hochschild (co)homology and the Kashiwara-Vergne conjecture.

Rationality and integrality in dynamical systems (Mike Zieve, Sept. 22, 2008):
I will present various results about the arithmetic of dynamical systems given by iterating a polynomial mapping over a ring. Sample topics include: describing the minimal N for which the backward orbit of a point under a given polynomial over a number field K contains infinitely many points of degree N over K; and determining the possible lengths of periodic and preperiodic forward orbits of a point under a polynomial mapping of a ring. I will also discuss connections with torsion in abelian varieties, Sen's theorem (Grothendieck's H^1 conjecture), and the Nottingham group.

Combinatorial identities and Hopf algebras (Earl Taft, October 20, 2008):
R. G. Larson and E. J. Taft showed that the space of linearly recursive sequences is a bialgebra. A coproduct formula for such a sequence can be interpreted as a quadratic identity on the coordinates of the sequence. This was extended by C. A. Futia, E. F. Mueller and E. J. Taft[CMT] to D-finite sequences. This means that from some point on, each coordinate is a linear combination of previous coordinates with variable(polynomial) coefficients. These D-finite sequences form a topological bialgebra, i.e., the coproduct is an infinite sum of tensor products of such sequences. Such a coproduct formula can still be interpreted as a quadratic identity on the coordinates, often of a combinatorial nature. In [FMT], we obtained such formulae and identities for the sequences (n!) and (n(n!)). Here we extend this to the sequences whose n-th term is ((n/k)(n!)) for each k=2, 3, 4,.... Here (n/k) is the binomial coefficient.

Infinite towers of cocompact lattices in Kac-Moody groups (Leigh Cobbs, November 3, 2008):
Let G be a locally compact Kac-Moody group of affine or hyperbolic type over a finite field Fq; G admits an action on its Tits building X. In the setting rank(G)=2, X is a locally finite, homogeneous tree. We can then use the combinatorial tools of Bass-Serre theory, namely graphs of groups, to construct discrete subgroups of G. We show that if q=2 then G contains a cocompact lattice Γ whose quotient Γ\X equals G\X, a simplex. We then give two distinct constructions of infinite towers

... Γ3 < Γ2 < Γ1 < Γ
of non-conjugate cocompact lattices in G. We give the graph of groups structure of these and other cocompact lattices, and discuss extensions of these infinite towers to rank-3 Kac-Moody groups using complexes of groups.

K-theory of some algebras associated to quivers (Guillermo Cortiñas, November 14, 2008):
Given a quiver Q and a field k, it is possible to associate several k-algebras. Best known among them is the path algebra, PQ. Localizing PQ one obtains a new algebra, the Leavitt algebra LQ. This algebra is equipped with an involution. If k is the field of complex numbers, LQ may be view as an algebra of operators in Hilbert space; its completion in the operator norm gives a C*-algebra, the Cuntz-Krieger algebra of the quiver. The topological K-theory of the Cuntz-Krieger algebra was computed in a now classical paper of Cuntz. In the talk we will discuss recent joint results with Pere Ara and Miquel Brustenga concerning the algebraic K-theory of LQ and its relation with the topological K-theory of the Cuntz-Krieger algebra.

Reduced K-theory of Azumaya algebras (Roozbeh Hazrat, December 1, 2008):

The theory of Azumaya algebras developed parallel to the theory of central simple algebras. However the latter are algebras over fields whereas the former are algebras over rings. One wonders how the K-theory of these objects compare to each other. We look at higher K-theory and reduced K-theory of these objects. We ask nice questions!

Spring 2008

W-algebras, quantum groups and combinatorial identities (Antun Milas, Feb. 5, 2008):
I will discuss a conjectural relationship between certain quantum W-algebras (vertex algebras) and finite-dimensional quantum groups associated to $sl_2$ (Hopf algebras). In the process we shall encounter interesting multisum identities.

The lattice of subfields of K(x) (Mike Zieve, Feb. 8, 2008:
I will present various results about the lattice of fields between K and K(x), where K is a field. These include classical results of Ritt, Schinzel, Fried, et al., as well as new results. I will also give some applications, for instance a recent joint result with Ghioca and Tucker describing all pairs of complex polynomials having orbits with infinite intersection.

Complex dynamics and shift automorphism groups (Zin Arai, Feb. 11, 2008):
Symbolic dynamics is a standard and powerful tool to understand chaotic dynamics. For example, we can identify the Julia set of quadratic polynomials with the one-sided shift space, the space of infinite sequences of 0 or 1, provided the parameter of the map is outside the Mandelbrot set. Furthermore, via the monodromy homomorphism, the topological structure of the Mandelbrot set is also captured by the automorphism group of the shift space.

In this talk, we consider the monodromy homomorphism for the complex Henon map, a 2-dimensional analog of the quadratic map. We need the shift space of bi-infinite sequences in this case, and the automorphism group of this space is much more complicated than that of the one-sided shift space. We propose a computer-assisted method to compute the monodromy homomorphism and show that automorphisms of the shift space can be used to determine the dynamics of the real Henon map.

Automorphism groups of curves (Mike Zieve, Feb. 25, 2008):
Hurwitz proved that a complex curve of genus g>1 has at most 84(g-1) automorphisms. In case equality holds, the automorphism group has a quite special structure. However, in a qualitative sense, all finite groups G behave the same way: the least g>1 for which G acts on a genus-g curve is on the order of (#G)*d(G), where d(G) is the minimal number of generators of G. I will present joint work with Bob Guralnick on the analogous question in positive characteristic. In this situation, certain special families of groups behave fundamentally differently from others. If we restrict to G-actions on curves with ordinary Jacobians, we obtain a precise description of the exceptional groups and curves.

Model categories versus derived categories (Chuck Weibel, March 3, 2008):
Quillen invented the notion of a model category in order to do homotopical algebra. We will consider these structures on the categories of R-modules, presheaves and sheaves, and show how localization works.

Rational points on homogeneous spaces (Parimala, March 10, 2008):
We discuss the following open concerning rational points on homogeneous spaces under connected linear algebraic groups. If a homogeneous space under a connected linear algebraic group has a zero cycle of degree one, does it admit a rational point? We explain the arithmetic case and some recent progress concerning this question for more general fields.

Formal differential representations, Faa di Bruno and the Riordan Group
(Tom Robinson, March 14, 2008):

First I will show explicitly how a calculation in Frenkel-Lepowsky-Meurman's book on vertex operator algebras, which I will in its essentials redo, can be viewed as an application of a formal representation of exponentiated derivations. The outcome of the calculation is Faa di Bruno's formula for the higher derivatives of a composite function. Then building on this result I will show how another application of an easy class of formal differential representation leads to the Riordan Group. No prerequisites necessary.

Real Groups and Topological Field Theory David Ben-Zvi, March 28, 2008:
I will explain current joint work with David Nadler, in which the representation theory of real reductive Lie groups is examined through the lens of topological field theory and the geometric Langlands program. Our main results show how to recover the representation theory of real forms of a complex group G from the representation theory of G, and how to deduce a Langlands dual description of the representation theory (a form of Soergel's conjecture, generalizing results of Vogan and Langlands).

The boson-fermion correspondence and one-sided quantum groups (Earl Taft, April 7, 2008):
Recent quantizations of the boson-fermion correspondence of classical physics use one half of the relations for the bialgebra of quantum matrices. Using this philosophy, A.Lauve, S. Rodriguez and myself have independently constructed certain one-sided qauntum groups, i.e., there is a left antipode which is not a right antipode. We will explain the connections between these two quantizations.

Plurisubharmonic functions on the octonionic plane and Spin(9)-invariant valuations on convex sets
(Semyon Alesker, April 21, 2008):

We introduce a class of plurisubharmonic functions on the octonionic plane O² and establish basic results about it. Then we apply these results to produce new examples of continuous valuatons on convex subsets of O²=R^{16}, in particular valuations invariant under the group Spin(9). The constructions use the determinant of octonionic hermitian matrices of size 2.

Witt vectors, Lambda-rings, and absolute algebraic geometry (Jim Borger, April 28, 2008):
I'll give an introduction to Witt vectors and Lambda-rings, and I'll explain how they're two different ways of looking at the same concept. Then I'll discuss how these give a "Lambda-equivariant" algebraic geometry, how it relates to usual algebraic geometry, and why one might care about it.

Subgroups of Algebraic Groups and Finite Groups (Richard Lyons, May 5, 2008):
We will discuss some similarities and differences between the subgroup structures of connected linear algebraic groups and finite groups.

Fall 2007

Conformal field theory and Schramm-Loewner evolution (Benjamin Doyon, Sept. 7, 2007):
The scaling limit of two-dimensional statistical models at criticality can be described by two theoretical frameworks: conformal field theory (that is, vertex operator algebras, their modules and representations), and Schramm-Loewner evolution (SLE). The first one has a long history, starting more than 20 years ago with works by both mathematicians and physicists, whereas the second one encompasses recent advances, starting in 2000 with a paper of Schramm until generalisations still under construction. The two frameworks seem quite unrelated in their formulation as well as in their applications. But it is nowadays believed by many that understanding the relation between them will allow us to make important steps in the understanding, both physical and mathematical, of critical regimes of statistical models. I will review the frameworks, advances made in relating them, and the many open problems. This talk will be accessible to non-specialists.

An introduction to open-closed conformal field theory (Liang Kong, Sept. 14, 2007):
Open-closed conformal field theory describes the perturbative open-closed string theory and some critical phenomena in condensed matter physics. It provides a powerful tool to study the still mysterious object called "D-brane", which is important to Kontsevich's homological mirror symmetry program. In this talk, I will outline a mathematical study of open-closed conformal field theory based on the theory of vertex operator algebra. In particular, I will give a tensor-categorical formulation of rational open-closed conformal field theory. I will also briefly discuss what D-branes are in our framework. This talk will be accessible to graduate students who know the definition of category.

Patching subfields of division algebras (Dan Krashen, Nov. 9, 2007):
There has been much work recently in understanding the structure of division algebras whose center is "2-dimensional." For example, in the case that the center is the function field of an algebraic surface, de Jong has shown that every such algebra has a cyclic maximal subfield. In this talk I will describe joint work with Harbater and Hartmann which uses the recent method of "field patching" (related to formal geometry) to understand all possible Galois groups of maximal subfields of division algebras over function fields of certain arithmetic surfaces.

Hopf and Lie algebras for renormalizable quantum field theories (Dirk Kreimer, Dec. 3, 2007):
Physicists have used the combinatorics of renormalization and the renormalization group routinely for a long time. The identification of the underlying algebraic structures in terms of Hopf and Lie algebras is more recent. We explain these algebras and their role in understanding Green functions in quantum field theory.

A new candidate for the nef cone of M0,n (Angela Gibney, Nov. 16, 2007):
There is a well known upper bound $F_{n}$ for the nef cone Nef$(\overline{M}_{0,n})$ of $\overline{M}_{0,n}$. The cone $F_{n}$ is an explicitly defined, polyhedral cone that contains Nef$(\overline{M}_{n})$. The F-conjecture asserts that Nef$(\overline{M}_{n})=F_{g,n}$. In this talk, I will describe a new candidate for the nef cone of $\overline{M}_{0,n}$. This is a polyhedral cone $C_{n}$ that Sean Keel, Diane Maclagan and I have proved is a sub cone of $F_{n}$. We can show that if $F_{n}$ were also contained in $C_{n}$, then it would imply that Nef$(\overline{M}_{0,n})=F_{n}=C_{n}$.

Spring 2007

Vertex operator algebras and recurrence relations (Bill Cook, March 30, 2007):
There are many important classes of examples of vertex operator algebras including Heisenberg VOAs, Virasoro VOAs, lattice VOAs, and the VOAs associated with affine Lie algebras.

We will begin with an introduction to the class of VOAs (along with their modules) associated with affine Lie algebras. Then in the latter part of the talk we will discuss an interesting theorem of Haisheng Li. Applying this theorem to our class of examples, we will obtain recurrence relations among the characters of these Vertex Operator Algebras (and VOA modules).

On a certain family of W-algebras (Antun Milas, April 7, 2007):
Rational conformal field theories can be characterized by the property that there are, up to equivalence, finitely many irreducible representations of the vertex operator algebra, and that every representation is completely reducible.

G-equvariant modular categories and Verlinde formula (Vincent Graziano, April 13, 2007):
Many features of a conformal field theory can be captured in the language of categories. Modular tensor categories provide the appropriate framework and we will start by discussing the properites of such a category. We will then introduce the Verlinde algebra associated to such a category, the action of the S-matrix, and the Verlinde formula.

Our goal will be to generalize this setup to the case of theories with additional symmetries, such as a vertex operator algebra with a finite group of symmetries. We discuss the extended Verlinde algebra, the S-matrix, and the 'extended' Verlinde formulas.

Vertex-algebraic structure of certain modules for affine Lie algebras underlying recursions (Corina Calinescu, April 20, 2007):
Many combinatorial identities and recursions have been proved or conjectured via vertex operator constructions of representations of affine Lie algebras.

In this talk we discuss vertex-algebraic structure of the principal subspaces of all the standard A1(1)-modules and we prove suitable presentations for these subspaces. These presentations were used by Capparelli, Lepowsky and Milas for the purpose of obtaining the classical Rogers-Ramanujan and Rogers-Selberg recursions. This is joint work with Jim Lepowsky and Antun Milas.

A Formal Variable Approach to Special Hyperbinomial Sequences (Tom Robinson, April 27, 2007):
In a nearly self-contained and elementary treatment, we develop the formal calculus used in the theory of vertex algebras to describe certain formal changes of variable. In particular, we extend the logarithmic formal Taylor theorem as found in the work of Y.Z. Huang, J. Lepowsky, and L. Zhang. We apply our results to obtain combinatorial identities concerning generalizations of the Stirling numbers and find that our development leads naturally to a combinatorial definition of the exponential Riordan group which was studied by L.W. Shapiro, S. Getu, W.J. Woan, and L.C. Woodson.

Fall 2006

Intertwining vertex operators and combinatorial representation theory (Corina Calinescu, Sept. 22, 2006):
In this talk we discuss vertex-algebraic structure of certain substructures, called principal subspaces, of standard modules for affine Lie algebras. We give suitable presentations of these subspaces and we derive Rogers-Ramanujan-type recursions satisfied by the graded dimensions of the principal subspaces. Part of the talk is based on joint work with Jim Lepowsky and Antun Milas. This talk will be introductory.

Certain generalizations of twisted affine Lie algebras and vertex algebras (Haisheng Li, Dec. 8, 2006):
We shall talk on certain generalizations of twisted affine Lie algebras and a natural connection of such Lie algebras with vertex algebras in terms of quasi modules.

Spring 2006

Vertex operators and sporadic groups (John Duncan, Jan. 20,2006):
In the 1980's, Frenkel, Lepowsky and Meurman demonstrated that the vertex operators of mathematical physics play a role in finite group theory by defining the notion of vertex operator algebra, and constructing an example whose full symmetry group is the largest sporadic simple group: the Monster. In this talk we describe an extension of this phenomenon by introducing the notion of enhanced vertex operator algebra, and constructing examples that realize other sporadic simple groups, including ones that are not involved in the Monster.

Solutions of families of polynomial equations (Jason Starr, January 27,2006):
Given a system of polynomials depending on parameters, when is there a polynomial map in the parameters whose output is a solution of the system for that choice of parameters? For 1-parameter systems, there is a polynomial map if for a general choice of the parameter every pair of solutions of the system can be connected by a 1-parameter family of solutions, i.e., if the variety is "rationally connected". I will discuss this theorem, the geometric interpretation and some consequences, and a conjecture for 2-parameter systems.

Congruence subgroup growth of arithmetic groups in positive characteristic (Balazs Szegedy, Feb. 10, 2006):
An arithmetic group, roughly speaking, consists of the integral points of a matrix group which is defined by polynomial equations. The most familiar example is SL(n,Z). The theory of arithmetic groups is an exciting meeting point of number theory, group theory, geometry, and combinatorics. We give a short introduction to the subject and present some recent results.

A smash product construction of nonlocal vertex algebras (Haisheng Li, Feb. 17, 2006):
We first introduce a notion of vertex bialgebra and a notion of module nonlocal vertex algebra for a vertex bialgebra. Then we present a smash product construction of nonlocal vertex algebras.

For every nonlocal vertex algebra $V$ satisfying a suitable condition, we construct a canonical bialgebra $B(V)$ such that primitive elements of $B(V)$ are essentially pseudo derivations and group-like elements are essentially pseudo endomorphisms. Furthermore, vertex algebras associated with Heisenberg Lie algebras as well as those associated with nondegenerate even lattices are reconstructed through smash products.

Chiral equivariant cohomology (Andy Linshaw, Feb. 24, 2006):
I will discuss a new cohomology theory that extends H. Cartan's cohomology theory of G* algebras. The latter is an algebraic abstraction of the topological equivariant cohomology theory for G-spaces, where G is a compact Lie group. Cartan's theory, discovered in the 50s and further developed by others in the 90s, gave a de Rham model for the topological equivariant cohomology, the same way ordinary de Rham theory does for singular cohomology in a geometric setting. The chiral equivariant cohomology takes values in a vertex algebra and includes Cartan's cohomology as a subalgebra. I will give a brief introduction to vertex algebras, and then discuss the construction of the new cohomology and some of the basic results and examples. This is a joint work with Bong Lian and Bailin Song.

Projective R[t]-modules and cdh cohomology (Chuck Weibel, March 31, 2006):
Let R be a finitely generated commutative algebra over a field of characteristic zero. Projective R-modules are classified by K0R and projective R[t]-modules are classified by K0R[t]. We prove that the quotient of these groups is a direct sum of R+/R and the cdh cohomology groups Hi(R,Ωi). This is joint work with Haesemeyer and Cortiñas.

Fall 2005

jdeg of finitely generated graded algebras and modules (Thuy Pham, Sept. 16, 2005):
Let R be a Noetherian ring, A be a finitely generated graded R-algebra where A=R[A_1] and let M be a graded A-module. We will assign to every finitely generated graded A-module M a new multiplicity, namely jdeg(M). This integer, which coincides with the classical multiplicity deg(M) when R is an Artinian local ring, captures various aspects of M besides its sheer size usually expressed in deg(M). In contrast to other extensions of deg(M), such as the arithmetic degree or the geometric degree, which require that R be a local ring, jdeg(M) places no such restrictions on R, it is truly a global object. I will describe some of its properties and applications.

Effective Hodge structures (Chuck Weibel, Sept. 23, 2005):
This is an introductory talk about Deligne's notion of Hodge Structures, and the more recent idea of effective Hodge Structures. Complex conjugation on the coordinates of the vector space V=Cn gives an involution, and a pure Hodge structure on V is a decomposition by subspaces Vp,q with Vp,q conjugate to Vq,p. It is effective if these only occur when p,q ≥0.

On certain principal subspaces of standard modules and vertex operator algebras (Corina Calinescu, Sept. 30, 2005):
Recently, S. Capparelli, J. Lepowsky and A. Milas initiated a new approach of getting Rogers-Ramanujan-type recursions by studying the principal subspaces of the standard sl(2)^-modules. We extend their approach to the untwisted affine Lie algebra sl(3)^. In this talk we give a complete list of relations for the principal subspaces of the standard sl(3)^-modules. Then, as a consequence of this result and vertex operator algebra techniques we obtain certain recursions. By solving them, we recover the graded dimensions (characters) of these principal subspaces.

Extensions of Rings and their Endomorphisms (Art, DuPre, Oct. 7, 2005):
Given rings I,Q we classify all rings R fitting into a short exact sequence     0 → I → R → Q → 0     of rings by means of cohomology classes. In the case of group extensions, it is necessary that the normal subgroup be abelian in order for the cohomology classes to form a group. However, because of the additive nature of the decomposition of a ring into cosets of an ideal, the cohomology classes form a group for arbitrary I. If R ≅ R1⊕R2 is a direct sum of rings, we may associate to any endomorphism f of R a 2x2 matrix
where fij:Rj→Ri are homomorphisms. We generalize this to the case where R is an arbitrary ring extension, determine the functional equations satisfied by the fij, and how such matrices multiply. We extend these results to the case where R carries a locally compact polish topology.

An isomorphism between two constructions of permutation-twisted modules for lattice vertex operator algebras (Katrina Barron, Oct. 14, 2005):
Twisted modules for vertex operator algebras arise in physics as the basic building blocks for "orbifold" conformal field theory, and arise in mathematics in the representation theory of infinite-dimensional Lie algebras. In this talk, we will consider two constructions of twisted modules in the case of the k-fold tensor product of a lattice vertex operator algebra with itself and a permutation automorphism acting on this tensor product.
One of these two constructions involves an operator based on the lattice, and the second involves an operator based on a coordinate transformation of the underlying conformal geometry modeled on propagating strings. However, by a theorem of the speaker, jointly with Dong, and Mason, they must produce isomorphic twisted modules. We construct an isomorphism explicitly thereby, from the point of view of physics, giving a direct link between the space-time geometry arising from the lattice and the conformal worldsheet geometry of propagating strings. This is joint work with James Lepowsky and Yi-Zhi Huang.

Kazhdan-Lusztig's tensor category and the compatibility condition (Lin Zhang, Oct. 21, 2005):
We study from the viewpoint of vertex operator algebras a braided tensor category of Kazhdan and Lusztig based on certain modules for an affine Lie algebra, by using a recent logarithmic generalization, due to Huang, Lepowsky and Zhang, of Huang and Lepowsky's tensor product theory for modules for a vertex operator algebra. We first give an equivalent form of the ``compatibility condition,'' one of the important tools in the theory of Huang and Lepowsky, in terms of a ``strong lower truncation condition.'' We use this to establish the equivalence of the two tensor product functors constructed in the two totally different approaches. Then, by using certain generalized Knizhnik-Zamolodchikov equations, we prove the ``convergence and expansion properties'' for this category and obtain a new construction of the braided tensor category structure. Compared to the original algebraic-geometric method, the vertex algebraic approach further establishes a vertex tensor category structure on this category.

Rational Maps on the Generic Riemann Surface (Bob Guralnick, Oct.28,2005):
Let X be the generic Riemann surface of genus g. If g > 6, Zariski proved that there is no solvable map from X to the Riemann sphere (i.e. a map with solvable monodromy group). We will discuss several generalizations and extensions of this result and some related open questions. Some of this is joint work with John Shareshian.

Leavitt path algebras (Gene Abrams, Nov 4, 2005):
Most of the rings one encounters as "basic examples" have what's known as the "Invariant Basis Number" property, namely, for every pair of positive integers m and n, if the free left R-modules RR(m) and RR(n) are isomorphic, then m=n. There are, however, large classes of rings which do not have this property. While at first glance such rings might seem pathological, in fact they arise quite naturally in a number of contexts (e.g. as endomorphism rings of infinite dimensional vector spaces), and possess a significant (perhaps surprising) amount of structure.

We describe a class of such rings, the (now-classical) Leavitt algebras, and then describe their recently developed generalizations, the Leavitt path algebras. One of the nice aspects of this subject is that pictorial representations (using graphs) of the algebras are readily available. In addition, there are strong connections between these algebraic structures and a class of C*-algebras, a connection which is currently the subject of great interest to both algebraists and analysts.

Supercategories and connections (Siddhartha Sahi, Nov 11, 2005):
We introduce the notion of a supercategory as a generalization of the tensor category of vector superspaces. We also define the concept of a "connection" in this context, and prove a series of extremely general quasi-isomorphism results generalizing the Harish-Chandra isomorphism.

A class of left quantum groups: Variation on the theme of SL_q(n) (Earl Taft, Dec 2, 2005):
For each n>1, we construct a left quantum group, which has the quantum special linear group SL_q(n) as homomorphic image. Whereas SL_q(n) is defined by quadratic relations plus the relation of degree n which sets the quantum determinant equal to 1, our left quantum group is defined by n^n relations of degree n, of which n! come from setting various versions of the quantum determinant equal to 1.(Joint work with Aaron Lauve).

Spring 2005

Generalizations of Tsen's theorem (Tom Graber, Feb 4, 2005):
Tsen's theorem is a classical result which says roughly that polynomials of low degrees in many variables with coefficients in the field of meromorphic functions on a compact Riemann surface always have solutions. I will describe joint work with Joe Harris, Barry Mazur, and Jason Starr which suggests that this result is best understood in connection with the geometry of rational curves.

Birkhoff decomposition in QFT and CBH formula (Li Guo, Feb 25, 2005):
We discuss the Hopf algebra approach of Connes and Kreimer to renormalization in pQFT, with emphasis on the role played by the Campbell-Baker-Hausdorff formula and Rota-Baxter operator in the Birkhoff decomposition of regularized characters. We also relate this decomposition to the factorization of formal exponetials by Barron-Huang-Lepowsky and the plus-minus decomposition for combinatiral Hopf algebras by Aguiar-Sottile.

Exotic Products of Linear Maps on Bialgebras (Earl Taft, March 4, 2005):
Linear maps on a bialgebra have two well-known associative products-composition and convolution. We define three more. Two are basically intertwining structures for the above two products. It is not clear if our third product is an intertwining structure. Our first two new products are related to certain generalized smash products. Applications will be given to left Hopf algebras, weak Hopf algebras and Hopf algebroids. (Joint work with E.H.Beggs, Univ. of Walews, Swansea)

Karoubi's construction for motivic cohomology operations (Zhaohu Nie, March 25, 2005):
Voevodsky constructed the reduced power operations in motivic cohomology following Steenrod's classical construction in topology. In this talk, I will present another construction of the motivic reduced power operations following a topological construction of Karoubi. The relation of the two constructions is, roughly speaking, that of a fixed point set and the associated homotopy fixed point set.

Uniqueness proof for Thompson's sporadic simple group (Gerhard Michler, april 1, 2005):
In 1976 J.G. Thompson announced the following Theorem: There is precisely one group E with the following properties: (a) All involutions of E are conjugate. (b) If z is an involution of E, H = C_G(z) and P = O_2(H), then P is extra-special of order 2^9 and H/P is isomorphic to the alternating group A_9.
Details of the proof for this result have never been published. In particular, the uniqueness question of the Thompson group Th had been considered to be an open problem by the experts until Weller, Previtali and the speaker have shown in 2003 that Th is uniquely determined up to isomorphism by a presentation of H. This presentation is due to Havas, Soicher and Wilson. It belongs to that sporadic simple group E which was originally discovered by Thompson and his collaborators at Cambridge. In the seminar I will outline our proof. Furthermore, I will mention some open problems related to Thompson's theorem.

Representations and Forms of Classical Lie algebras over finite fields (Bin Shu, April 8, 2005):
By introducing Frobenius-Lie morphism, a connection between finite-dimensional representations of finite Lie algebras over finite fields and their algebraic closures is established, which enables us to understand irreducible representations of classical Lie algebras over a finite field $F_q$ through the ones of its extension over $\bar F_q$. Moreover, Frobenius-Lie morphisms provide us an approach to the determination of the number of forms of classical Lie algebras, which is different from the method used in "Modular Lie Algebras", by G.B. Seligman. This work is done jointly with Jie Du.

Infinitesimal bialgebras and associative classical Yang-Baxter equations (Kurusch Ebrahimi-Fard, April 15, 2005):
Infinitesimal bialgebras are generalized bialgebras with a comultiplication that is not an algebra homomorphism, but a derivation. They were introduced by Joni and Rota (Stud. Appl. Math. 61 (1979), no. 2, 93-139). M. Aguiar developed a theory for these objects analogous that of ordinary Hopf algebras, showed their intimate link to Rota-Baxter algebras, Loday's dendriform algebras, and introduced the associative classical Yang-Baxter equation. In this talk we will briefly review and generalize the above setting. Also, we will explore the factorization theorems related to Rota-Baxter algebras and the BCH-formula in this context.

Koszul duality and posets (Bruno Vallette, April 21, 2005):
Associated to any operad, we define a poset of partitions. We prove that the operad is Koszul if and only if the poset is Cohen-Macaulay. In one hand, this characterisation allows us to compute the homology of the poset. This homology is given by the Koszul dual cooperad. On the other hand, we get new methods for proving that an operad is Koszul.

Relative twisted vertex operators associated with the roots of the Lie algebras A_{1} and A_{2}   (Christiano Husu, April 29, 2005):
The Jacobi identity for vertex operator algebras incorporates a family of "cross-brackets," including the Lie bracket, and expresses these brackets as the product of an "iterate" of vertex operators with a suitable form of the formal delta function. The generalization of the Jacobi identity to relative vertex operators requires the introduction of "correction factors" which preserve the vertex operator structure of the Jacobi identity. These correction factors, in turn, uncover the main features of Z-algebras (generalized commutator and anti-commutator relations) in the computation of a residue of the relative (twisted) Jacobi identity.

More specifically, using k copies of the weight lattices of the Lie algebras A_{1} and A_{2} in the diagonal embedding, we construct relative twisted vertex operators equivalent to Z-algebra operators that determine the structure of standard A_{1}(1) and A_{2}(2)-modules. Applying the properties of the delta function, the corresponding generalized commutator and anti-commutator relations appear as residues of the Jacobi identity for relative twisted vertex operators.

In this talk we will introduce the notion of partial actions of groups on algebras in a pure algebraic context. Partial skew group rings and partial skew polynomial rings will be defined. We will discuss the associativity question and some other related problems.

Fall 2004

Conformal field algebras and tensor categories (Liang Kong, Oct. 1, 2004):
Conformal field theories have both holomorphic and antiholomorphic parts, which are sometimes called chiral conformal field theories. In genus-zero and genus-one cases, chiral conformal field theories have been constructed from a general class of vertex operator algebras and their representations, and in general these theories have monodromies. To construct conformal field theories without monodromies, we need to put chiral theories together to cancel the monodromies. In genus-zero, such conformal field theories are described by what we call "conformal field algebras." In this talk, we will discussion the notion of conformal field algebra, their relation with algebras in tensor categories, and a construction of such algebras.

Cherednik and Hecke algebras of orbifolds (Pavel Etingof, Oct. 15, 2004):
The rational Cherednik algebra is attached to a finite group G acting on a vector space V, i.e., to the orbifold V/G. I will explain how the theory of Cherednik algebras can be extended to an arbitrary orbifold (algebraic or complex analytic), and how to define the KZ functor for such algebras.
This leads to a construction of a flat deformation of the group algebra of the orbifold fundamental group of any complex orbifold Y whose universal cover has a finite second homotopy group. These deformations include all known Hecke algebras (usual, complex reflection, affine, double affine). The talk is based on my paper math.QA/0406499.

Modular theory, quantum subgroups and quantum field theory (Adrian Ocneanu, Oct. 29, 2004):
We describe the connections between modular invariants, topological quantum doubles and the construction and classification of quantum subgroups. We discuss applications to quantum field theoretical models.

Vertex operator coalgebras: Their operadic motivation and concrete constructions (Keith Hubbard, Nov. 5, 2004):
Arising from the study of conformal field theory, vertex operator coalgebras model the surface swept out in space-time as a closed string splits into two or more strings. By studying the theory of operads, a structure introduced by May to study iterated loop spaces, the structure of both vertex operator algebras and vertex operator coalgebras may be developed.

This talk will define the notion of operad, show how operads geometrically motivate associative algebras and coassociative coalgebras, and then analogously use operads to motivate vertex operator algebras and vertex operator coalgebras. The talk will conclude with examples of vertex operator coalgebras that are constructed via vertex operator algebras with appropriate bilinear forms.

Homotopy theory for Motives (Charles Weibel, Nov. 12, 2004):
An introduction to the Morel-Voevodsky construction of homotopy theory for algebraic varieties which underlies modern notions of motives. The idea is that a "space" should be a jazzed-up object built up out of varieties using simple constructions like quotients, and that the affine line should play the role of the unit interval.

The Van Est spectral sequences for Hopf algebras (Edwin Beggs, Nov. 19, 2004):

In classical geometry there have been results about the cohomology of manifolds with Lie group actions, and the relation between the topological cohomology of the group and its Lie algebra cohomology, for about 50 years. I shall give noncommutative analogues of some of these results, in terms of Hopf algebras acting on algebras with differential structure. I shall begin with a brief review of noncommutative differential geometry and de-Rham cohomology.

Quasi-Hoph algebras, twisting and the KZ equation (Edwin Beggs, Dec. 10, 2004):

In this informal talk I'll give the definition of quasi-Hopf algebras, some examples (and some conjectural examples) of twisting, including the Knizhnik-Zamolodchikov (KZ) equation.

Spring 2004

Toric Hilbert schemes (Diane Maclagan, Jan 26, 2004):
Toric Hilbert schemes have broad connections to other areas of mathematics, including optimization, geometric combinatorics, algebraic geometry, and representations of finite groups and quivers. They parameterize all ideals in a a polynomial ring with the simplest possible multigraded Hilbert function. I will introduce these objects, and discuss some of the applications.

Orbifold Cohomology of Toric Stacks (Greg Smith, Jan 28, 2004):
Quotients of a smooth variety by a group play an important role in algebraic geometry. In this talk, I will describe an interesting collection of quotient spaces (called toric stacks) defined by combinatorial data. As an application, I will relate the orbifold cohomology of a toric stack with a resolution of the underlying singular variety.

On Deformation Quantization in Algebraic Geometry (Amnon Yekutieli, March 12, 2004):
We study deformation quantization of Poisson algebraic varieties. Using the universal deformation formulas of Kontsevich, and an algebro-geometric approach to the bundle of formal coordinate systems over a smooth variety X, we prove existence of deformation quantization of the sheaf of functions OX (assuming the vanishing of certain cohomologies). Under slightly stronger assumptions we can classify all such deformations.

Conformal algebras and their representations (Alexander Retakh, March 26, 2004):
Conformal algebras first appeared as an attempt to provide algebraic formalism for conformal field theory (as part of the theory of vertex algebras). They are also closely related to Hamiltonians in the formal calculus of variations.

In this talk, however, I will present conformal algebras as a self-contained theory and will mostly concentrate on their representations, in particular, on the conformal analogs of matrix algebras. These objects are related to certain subalgebras of the Weyl algebra and the algebra gl{\infty}.

Capture the flag: towards a universal noncommutative flag variety (Aaron Lauve, April 2, 2004):
The standard way to build flag algebras from a set of flags is to use the determinant to coordinatize the latter (then the former is just the polynomial algebra in the coordinate functions for these coordinates). There is a perfectly reasonable notion of noncommutative flags, but what are we to do about the lack of a determinant in noncommutative settings? In this talk I will: (1) use the Gelfand-Retakh quasideterminant to build a generic noncommutative Grassmannian algebra, (2) specialize this generic Grassmannian to recover the well-known Taft-Towber quantum Grassmannian, (3) explain what steps are left before we can build a generic flag algebra. This talk should be accessible to first and second year graduate students.

The affine algebra A22 and combinatorial identities (Stefano Capparelli, April 9, 2004):
I will give a brief outline of the Lepowsky-Wilson Z-algebra approach to classical combinatorial identities and the Meurman-Primc proof of the generalized Rogers-Ramanujan identities. I will next outline the application of this theory to the construction of the level 3 standard modules for the affine algebra A22 and the corresponding combinatorial identities as well as Andrews' combinatorial proof of these identities. I will discuss some current ideas for a possible approach to these identities and their generalizations using intertwining operators. Finally, I will mention the apparent link between level 5 and 7 standard modules for the affine algebra A22 and some other Rogers-Ramanujan-type identities of Hirschhorn.

Extremal simplicial polytopes (Uwe Nagel, April 16, 2004):
In 1980 Billera-Lee and Stanley characterized the possible numbers of i-dimensional faces of a simplicial polytope. Its graded Betti numbers are finer invariants though little is known about them. However, among the simplicial polytopes with fixed numbers of faces in every dimension there is always one with maximal graded Betti numbers. In the talk, this result will be related to the more general problem of characterizing the possible Hilbert functions and graded Betti numbers of graded Gorenstein algebras and key ideas of its proof will be discussed.

Dendriform algebras and linear operators (Li Guo, April 23, 2004):
Dendriform algebras refer to a class of algebra structures introduced by Loday in 1996 with motivation from algebraic K-theory. The field has expanded quite much during the last couple of years, with connections to operad theory, math physics, Hopf algebras and combinatorics. A recent observation is that some basic dendriform algebras are induced by linear operators, such as Baxter and Nijenhuis operators, and more complicated such algebras can be decomposed as products in operad theory. We will discuss these developments.

There exists a one-sided quantum group (Earl Taft, April 30, 2004):
Bialgebras with a left antipode but no right antipode were constructed in the 1980's by J.A.Green, W.D.Nichols and E.J.Taft. Recently, S.Rodriguez-Romo and E.J.Taft tried to construct such a one-sided Hopf algebra within the framework of quantum groups, starting with roughly half the defining relations for quantum GL(2). Asking that the left antipode constructed be an algebra antimorphism led to some additional relations, but the result was a new(two-sided) Hopf algebra. Now we start with roughly half the relations for quantum SL(2) but ask that our left antipode constructed reverse order only on irreducible monomials in the generators. The result is a quantum group with a left antipode but no right antipode.


Constructing tensor categories from from finite groups (Edwin Beggs, Sept 12, 2003):
First we consider the algebra structure induced on a set of coset representatives of a subgroup of a finite group. Associated to it is a non-trivial tensor category, which we construct. There is an algebra in this category whose representations consist of the entire category.

If we apply a double construction to this, we arrive at a braided category and a braided Hopf algebra. It turns out that this is a ribbon category, and (at least sometimes) a monoidal category.

Open-string vertex algebras (Liang Kong, Oct. 3, 2003):
This is joint work with Y.-Z. Huang.
We introduce notions of open-string vertex algebra, conformal open-string vertex algebra and variants of these notions. These are "open-string-theoretic," "noncommutative" generalizations of the notions of vertex algebra and of conformal vertex algebra. Given an open-string vertex algebra, we show that there exists a vertex algebra, which we call the "meromorphic center" inside the original algebra such that the original algebra yields a module and also an intertwining operator for the meromorphic center. This result gives us a general method for constructing open-string vertex algebras. Besides obvious examples obtained from associative algebras and vertex (super)algebras, we give a nontrivial example constructed from the minimal model of central charge c = 1/2 . We also discuss the relationship between the gradingrestricted conformal open-string vertex algebras and the associative algebras in braided tensor categories. We also discuss a geometric and operadic formulation of the notion of such algebra and the relationship between such algebras and a so-called Swiss-cheese partial operad.

The Development of Standard Monomial Theory (C. Musili, Oct. 10, 2003):
The main phases of the development of Standard Monomial Theory (SMT) and some of its applications to Geometry and Commutative Algebra will be surveyed without assuming anything and, more importantly, without becoming technical.

Let G be a semi-simple, connected and simply connected algebraic group, defined over an algebraically closed field of characteristic 0. Fix a maximal torus T, a Borel subgroup B containing T, and a maximal parabolic subgroup P containing B. Fix also the root system of G relative to T, the positive/simple roots relative to B, etc. Let W = W(G) be the Weyl group of G.

Let V be a fundamental representation of G corresponding to P. The first main aim of SMT is to construct a "nice" basis for each of the T-weight subspaces in V, having some "compatibility" properties with that of the extremal weight spaces and satisfying some "geometric" properties, etc.

Let M be an irreducible representation of G and express it as a subquotient of the appropriate tensor product of suitable fundamental representations of G. The second main aim of SMT is to construct bases for the weight spaces of M in terms of those constructed for the fundamental representations.

The Motivic DGA (Roy Joshua, Oct. 17, 2003):
We will outline the structure of an E_{\infinity} algebra on the motivic complex drawing the parallel with the singular complex where such a structure was provided by Hinich and Schechtman. We will also consider some applications like the construction of a category of relative Tate motives for a large class of varieties and the construction of cohomology operations in motivic cohomology with finite coefficients. (This is joint work with Peter May.)

Modules, Comodules, Entwinings and Braidings (Bodo Pareigis, Oct. 24, 2003):

  • The category of A-modules and the underlying forgetful functor to vector spaces determine the algebra A up to isomorphism. A forgetful functor from A-modules to B-modules defines a unique algebra homomorphism f: B -> A.

    We will study certain A-modules that are also C-comodules, called entwined (A,C)-modules. This involves a certain double action-coaction t , an entwined structure, between A and C. Similar techniques as above allow to reconstruct A and C plus the entwined structure t from the category of entwined (A,C)-modules and the forgetful functor to vector spaces.

    There have been some (failed) attempts to find the correct definition of homomorphisms between entwined algebra-coalgebra structures (A,C,t) -> (A',C',t'). We will give the correct definition using the approach given above, which lead to certain measurings and comeasuring. We will show how these techniques can be used to get other interesting results about entwined structures.

    A vertex operator algebra approach to the construction of a tensor category of Kazhdan-Lusztig (Lin Zhang, Nov. 21, 2003):
    In contrast to the ordinary tensor product of modules for a Lie algebra, the known construction of the tensor product of modules of a *fixed* level for an affine Lie algebra is completely nontrivial and in essence uses ideas from conformal field theory. As a payoff it produces a braided tensor category structure. I will first summarize Kazhdan-Lusztig's construction of this tensor product, and then, using recent joint work with Huang and Lepowsky, I will show how this can be incorporated into vertex operator algebra theory and how the braided tensor category can be proved to in fact have a "vertex tensor category" structure. I will also present another application of this work that is related to the module category for vertex algebras associated with hyperbolic even lattices.

    Finite extensions of vertex algebras   (Victor Ostrik, Dec. 5, 2003):
    In this talk we will discuss the problem of classification of all extensions $V\subset V'$ of a given vertex algebra $V$ such that $V'$ is a finite length module over $V$. Under certain assumptions on the algebra $V$ this problem is equivalent to the classification of commutative algebras in the tensor category of $V-$modules (Huang-Kirillov-Lepowsky). We review what is known about the latter problem, in particular known classification results for affine Lie algebras on positive integer level and for holomorphic orbifolds.

    Orbifolding the chiral de Rham complex   (Matthew Szczesny, Dec. 12, 2003):
    Given a smooth variety X with the action of a finite group G, we construct twisted sectors for the chiral de Rham complex from sheaves of twisted modules supported along fixed point sets. The BRST cohomology of the twisted sectors is isomorphic to the Chen-Ruan orbifold cohomology of of the orbifold [X/G], and the partition function yields the orbifold elliptic genus.

    The pure mapping class group of a Cantor set (Frederick Gardiner, May 2, 2003):
    This is joint work with Nikola Lakic.
    It is shown that the pure mapping class group of the complement in the complex plane of the standard middle-thirds Cantor set acts discretely on the Teichmuller space of Cantor sets of bounded geometric type.

    Affinization of commutative algebras (Michael Roitman, April 25, 2003):
    If V = V_0 + V_1 + V_2 + ... is a vertex algebra graded so that dim V_0 = 1 and V_1 = 0, then V_2 has a structure of commutative (but non-associative in general) algebra with an invariant bilinear form. We show that for any commutative algebra A with a non-degenerate invariant bilinear form there is a vertex algebra V, graded as above, such that A = V_2. Moreover, if A has a unit e, then V can be chosen so that 2e is a Virasoro element.

    Twisted K theory from the Dirac spectral flow (Constantin Teleman, April 18, 2003):
    Based on the archetypal example of spectral flow over the circle, described by Atiyah and Singer, we describe an explicit construction of twisted K theory classes over a compact Lie group G, equivariant for the adjoint action, from positive energy representations of the loop group. The construction involves a family of Dirac-Ramond operators (studied by Taubes, Kostant and Landweber) and provides an inverse of the "kirillov correspondence", which assigns unitary representations to certain co-adjoint orbits.

    Tensor category theory for modules for a vertex operator algebra -- introduction and generalization (Lin Zhang, April 11, 2003):
    There is a well-developed tensor category theory for certain modules of a fixed positive integral level for an affine Lie algebra, important in the study of conformal field theory. Two mathematical constructions are available: one by Finkelberg following Kazhdan-Lusztig's work on a related category, using algebro-geometric methods, and the other by Huang and Lepowsky using vertex operator algebra theory, which works for a very large class of categories of modules for rational vertex operator algebras. In this talk I will give an introduction to the vertex algebraic approach, and explain how, by using logarithmic intertwining operators, this can be applied to the category considered by Kazhdan and Lusztig. I will also explain the relation of the two constructions of this category. No tehnical background will be assumed in this talk.

    Schiffer Variation in Teichmüller Space and Determinant Line Bundles (David Radnell, March 28, 2003):
    A construction of conformal field theories, in the sense of Segal and Kontsevich, from vertex operator algebras is essentially complete in genus zero and one. However, in higher genus, some basic analytic and geometric problems must be solved in order to even formulate some fundamental structures, such as modular functors and holomorphic weakly conformal field theories in the sense of Segal.
       The basic geometric objects are Riemann surfaces with analytically parametrized boundaries and their associated determinant lines. The formulations of modular functors and holomorphic weakly conformal field theories are based on the highly nontrivial assumptions that the moduli space of such surfaces is an infinite-dimensional complex manifold, the determinant lines form a holomorphic line bundle over this moduli space and that the sewing operation is holomorphic. I will outline a proof of these results using deep classical results from Teichmüller theory and Schiffer variation.
    No background in geometry will be assumed.

    Novikov algebras and vertex (operator) algebras (Chengming Bai, March 14, 2003):
    Novikov algebras were introduced first by I.M. Gelfand and I.Y. Dorfman in connection with Hamiltonian operators in the formal variational calculus. Later they were also introduced to study the Poisson brackets of hydrodynamic type by S.P. Novikov et al. The finite-dimensional Novikov algebras can induce an interesting class of the infinite-dimensional Virasoro type Lie algebras. Hence they correspond to a class of vertex (operator) algebras. I will give a brief survey of the study of finite-dimensional Novikov algebras and the relations between them and their corresponding vertex (operator) algebras.

    Lie algebras associated with derivation-simple algebras (Yucai Su, March 7, 2003):
    We first classify the pairs (A,D), where A is a commutative associative algebra with an identity element over an algebraically closed field F of characteristic zero, D is a finite dimensional F-vector space consisting of commuting locally finite derivations of A. Then using these pairs, we construct some in general not finitely graded Lie algebras of generalized Cartan type, and study their structure theory and representation theory.


  • "Modular group action in the center of the small quantum group" (Anna Lachowska, Dec.6, 2002):
    The small quantum group ul was introduced by Lusztig as a certain finite dimensional Hopf algebra associated to a semisimple complex Lie algebra g and a primitive (complex) l-th root of unity q. According to Lyubashenko and Majid, in many cases ul admits a bijective action of two operators obeying the modular identities. This action stabilizes the center Z of ul and can be used to study its structure.
          We will consider the smallest modular-invariant subspace in Z which contains the obvious central elements (the Harish-Chandra center). This subspace is a subalgebra in Z roughly twice bigger than the Harish-Chandra center, and it coincides with the whole center Z in case g = sl_2. It also contains a nilpotent modular-invariant ideal, which admits an interesting representation-theoretical interpretation similar to the Verlinde algebra of the fusion category.

  • "Support Varieties for Finite Group Schemes" (Julia Pevtsova, November 22, 2002):
    This is joint work with Eric Friedlander.
    To each finite-dimensional module M of a finite group scheme G (i.e. finite dimensional cocommutative Hopf algebra over a field of positive characteristic) we associate a geometric construction of a ``representation-theoretic support space'' of M. Our construction specializes to two seemingly different constructions in modular representation theory: Carlson's rank varieties for elementary abelian p-groups and representation-theoretic support varieties of restricted Lie algebras. We further exhibit a natural homeomorphism from the representation-theoretic support space of the trivial module to the projectivization of the spectrum of the cohomology algebra of G. For every finite dimensional G-module M, this homeomorphism restricts to a homeomorphism between the representations-theoretic support space and the projectivization of the cohomological support variety of M.

  • "Twisted vertex operator algebra modules and Bernoulli polynomials" (Benjamin Doyon, November 1, 2002):
      In the construction of twisted modules for vertex operator algebras, one can define twisted vertex operators by normal-ordered products of more ``basic'' twisted vertex operators. One also needs to introduce a certain subtle formal operator in the construction; this gives, in particular, a correction term for the action of the zero mode of the Virasoro algebra on a twisted module. The generalization of this term to the case of a central extension of an algebra of differential operators of higher order is a very non-trivial problem from this point of view.
          We start from the twisted Jacobi identity and derive various ``commutativity'' and ``associativity'' relations. They allow us to define twisted vertex operators from more basic ones without explicit reference to an extra formal operator, and to calculate correction terms more conceptually. Bernoulli polynomials appear when we use ``cylindrical coordinates'', where, as shown by J. Lepowsky, the algebra simplifies drastically.
    This is joint work with J. Lepowsky and A. Milas.

  • "Explicit norm one elements for ring actions of finite abelian groups" (Christian Kassel, October 25, 2002):
    It is known that the norm map NG for the action of a finite group G on a ring R is surjective if and only if for every elementary abelian subgroup U of G the norm map NU is surjective. Equivalently, there exists an element xG in R satisfying NG(xG) = 1 if and only for every elementary abelian subgroup U there exists an element xU in R such that NU(xU) = 1. When the ring R is noncommutative, it is an open problem to find an explicit formula for xG in terms of the elements xU. E. Aljadeff and the speaker solved this problem when the group G is abelian.

  • "Is there a one-sided quantum group?" (Earl Taft, October 18, 2002):
    In the 1980's, J.A. Green, W.D.Nichols and EJT constructed a left Hopf algebra, i.e., a bialgebra with a linear map S satisfying the left antipode condition, but not the right one. It has a freeness feature that places it outside the realm of quantum groups. Recently, S. Rodriguez-Romo and EJT tried to construct a left Hopf algebra in the world of quantum groups. We did not yet succeed, but the effort led to some new quantum groups, modeled partially on quantum GL(2), with the peculiar property that they remain noncommutative when q=1 [ Letters in Mathematical Physics 61 (2002), 41-50.] We are trying to modify our procedures, with the hope of finding a left quantum group. If such a thing exists, it might reflect some lack of symmetry of interest to physicists.

  • "Raga Bhimpalasi: The Vaserstein-Suslin Jugalbandhi" (Ravi Rao, Oct. 11, 2002):
    The study of unimodular rows, and their orbit spaces, over a commutative ring with 1, lies in the fertile cross-section of ideas from Algebra, Algebraic Topology, Number Theory, and Algebraic Geometry. Witt group structures, Cohomotopy groups, Mennicke symbols, Reciprocity Laws, etc. make their appearance very naturally.
       We shall discuss the connection of the study of orbit spaces, via a symbiosis of constructions of L.N. Vaserstein, A. Suslin, and its relation to problems in classical K-theory, and to the program of J.-P. Serre, which interconnected the study of projective R-modules with problems of efficient generation of ideals of R.

  • "Conformal field theory and vertex operator algebras" (Matthias Gaberdiel, Sept.27, 2002):
    I plan to give an informal introduction into conformal field theory and its relation to vertex operator algebras. Towards the end I shall also discuss the so-called C_2 condition of Zhu and some of its implications.

  • "Differential equations, duality and modular invariance" (YZ Huang, Sept.20, 2002):
    I will explain a result obtained recently on genus-one conformal field theories. Let V be a vertex operator algebra satisfying the C_2-cofiniteness condition and certain finite reductive properties. Then the q-traces of products of geometrically-modified intertwining operators are shown to satisfy systems of differential equations which can be chosen to be regular at any given possible singular point. Genus-one correlation functions are constructed as the analytic extensions of these q-traces. We prove duality properties for these genus-one correlation functions, including commutativity and associativity. Using the associativity property and the modular invariance for one-point functions, we establish the modular invariance for genus-one correlation functions. I will start with the definition of conformal field theory and will explain briefly the notion of vertex operator algebra.

  • "Differential equations and intertwining operators" (YZ Huang, May 3, 2002):
    In the conformal field theories associated to affine Lie algebras (the Wess-Zumino-Novikov-Witten models) and to Virasoro algebras (the minimal models), the Knizhnik-Zamolodchikov equations and the Belavin-Polyakov-Zamolodchikov equations, respectively, play a fundamental role. Many important results (for example, the constructions of braided tensor category structures and intertwining operator algebras) for these theories are obtained using these equations.
       In this talk, I will explain a recent result which establishes the existence of certain different equations of regular singular points satisfied by products and iterates of intertwining operators for a vertex operator algebra whose modules satisfy a certain finiteness condition. Immediate applications of these equations are a construction of braided tensor categories on the category of modules for the vertex operator algebra and a construction of intertwining operator algebras (or chiral genus-zero conformal field theories) from irreducible modules for the vertex operator algebra.

  • "A simple nil ring exists" (Smoktunowicz, April 19, 2002):
    Over 40 years ago, a simple radical ring was constructed by Sasiada. It remained an open question whether a simple nil ring exists. (Nil means that every element is nilpotent.) We construct a simple nil ring over any countable field. We will describe this construction, and also mention several open questions such as: Is there a simple nil ring over an uncountable field?

  • "Once again about Bethe Ansatz" (Lukyanov, March 29, 2002):
    We shall discuss an intriguing relation between roots of the Bethe ansatz equations corresponding to vacuum states of the $XXZ$ spin chain and the spectrum of one-dimensional Scr${\ddot {\rm o}}$dinger operator with homogeneous potential.


  • "Congruence subgroups of SL2(Z[1/s]), after Serre" (Weibel, Dec.7, 2001):
    The Congruence Subgroup Problem for any matrix group G is to determine if every subgroup of finite index is a congruence subgroup, and if not to describe the obstruction. If G is SL_n(Z[1/s]) and n>2, the Euclidean Algorithm easily yields a positive solution. Less well known is the case of 2x2 matrices.
    Following Serre, we will show that the answer is 'no' for Z, and Z[1/p], but 'yes' for Z[1/s] otherwise.
    This is intended to be an expository talk. One motivation is that subgroups of finite index in SL_2(Z) play an important role in number theory.

  • "Certain noncommutative analogues of vertex algebras" (Haisheng Li, Nov.30, 2001):
    We define and study a certain noncommutative analogue of the notion of vertex algebra. We show how to construct such algebras by using a set of compatible weak vertex operators.

  • "Vertex operator algebras and conformal field theories III" (YZ Huang, Nov.16, 2001):
    I will continue the discussions on modular functors and weakly conformal field theories and on the consequences of the existence of such theories, including the Verlinde formula. I will also briefly explain Segal's idea on how to obtain real conformal field theories from rational weakly conformal field theories. At the end, the existing results and open problems will be discussed.

  • "Some biparametric examples of Quantum Groups" (Parashar, Nov. 9, 2001):
    I will present simple examples of the standard and nonstandard (or Jordanian) quantum groups as well as their biparametric versions. The scheme is then extended in the wider context of the corresponding `coloured' counterparts. Within the framework of the R-matrix approach, I will also discuss some basic algebraic and geometric results from the theory of coloured quantum groups and outline possible physical and mathematical applications.

  • "Vertex operator algebras and conformal field theories II" (YZ Huang, Nov.3, 2001):
    This talk is a continuation of my talk on October 12. I will explain how to construct genus-zero conformal field theories from vertex operator algebras and why we need modules and intertwining operators when we want to construct maps associated to genus-one surfaces. I will also discuss weakly-conformal field theories and consequences of the existence of such theories, including the Verlinde formula.

  • "Elliptic curves and quantum tori" (Soibelman, Oct.26, 2001):
    I plan to discuss the program of non-commutative compactifications I suggested two years ago. The main example will be non-commutative degenerations of elliptic curves. I will explain why quantum tori appear on the boundary of the moduli space of elliptic curves. I will discuss the relations to algebraic and symplectic geometry, q-difference equations, etc. The talk will consist largely of conjectures and speculations.

  • "Vertex operator algebras and conformal field theories" (YZ Huang, Oct.12, 2001):
    Conformal field theories were defined mathematically around 1987 by Kontsevich and Segal in terms of properties of path integrals. A construction of such a theory can be viewed in a certain sense as a construction of certain path integrals. However, up to now, there is still no complete published construction of examples of conformal field theories satisfying this definition.

    On the other hand, around 1986, a notion of vertex operator algebra was introduced and studied in connection with the representation theory of infinite-dimensional Lie algebras and the Monster by Borcherds and Frenkel-Lepowsky-Meurman. Since then, the theory of vertex operator algebras has been developed rapidly and has found applications in a number of branches of mathematics. In this series of talks, I will explain a research program to construct conformal field theories in the sense of Kontsevich and Segal. Both the existing results and unsolved problems will be discussed.

  • "Effective Representation Theory of Finitely Presented Algebras" (Letzter, Sept.28, 2001):
    Let n be a positive integer, and let R be a finitely presented algebra over a field k. Consider the following questions: Does R have an irreducible n-dimensional (over k) representation? How many irreducible n-dimensional representations does R have? Is every n-dimensional representation of R semisimple? In this talk I will discuss algorithmic approaches to answering these questions.

  • "Hilbert functions of non-standard bigraded algebras" (Trung, Sept.21, 2001):
    The talk concerns bigraded algebras generated by elements of bidegrees (1,0) and (a,1) for different non-negative integers a. We show that the Hilbert function of such a bigraded algebra is equal to a polynomial for certain range. Moreover, the total degree and the degree of this polynomial in the first variable can be expressed in terms of the dimension of certain quotient algebras. These results cover recent results of P. Roberts on the existence of Hilbert polynomial in the case the bigraded algebra is generated by elements of bidegrees (1,0), (0,1), (1,1) which are related to Serre's positivity conjecture on intersection multiplicities. Moreover, these results can be applied to study diagonal subalgebras of bigraded Rees algebras.

  • "On a q-analog of the McKay correspondence" (Kirillov, Sept.14, 2001):
    It is well known that finite subgroups in SU(2) are classified by simply-laced affine Dynkin diagrams, i.e., affine ADE diagrams. This calssification, known as McKay correspondence, is one of the many related ADE-type classifications (e.g., it is related with ADE classification in singularity theorey). In this talk, we give an analogue of this result for the quantum group U_q sl(2) with q beig a root of unity. This turns out to be related with the classification of modular invariants in Conformal field theory based on integrable representations of affine sl(2).

  • "A geometric approach to elliptic cohomology": (Igor Kriz/ Feb.16, 2001)
    This talk will discuss the current state of the speaker's project of constructing a geometric model of elliptic cohomology. This proposed construction is related to theta functions and vertex operator algebras. The talk will describe the construction, and the links between its conjectured topological properties and their algebraic and geometric counterparts.

  • "Functors with transfer": (C. Weibel/ Feb.2, 2001)
    Many of the ideas coming out of Motivic Cohomology yield new ideas and questions when translated into commutative ring theory. We will describe some of these techniques and apply them to questions about projective modules.

    Charles Weibel / weibel @ / January 1, 2018