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

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

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 sl_{2m}" 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 ofp-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

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

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

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

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"

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"

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"

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"

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

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.

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

(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)

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)

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.

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).

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.

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.

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

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 M_{0,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.

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

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 Lectures31 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.

9 Sept Colonel Henry Rutgers -------- Department Reception ---------------- 16 Sept: Thuy Pham Rutgers "jdegof 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).

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.

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 CourantD'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)

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 A_{2}^{2}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)

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)

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)

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"

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).

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)

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)

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"

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"

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"

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"

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)"

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" 1 May: Toma Albu U.Wisc.-Milwaukee "GLOBAL KRULL DIMENSION AND GLOBAL DUAL KRULL DIMENSION OF RINGS"

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"

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"

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

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"

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"

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"

**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 SchÃ¼tzenberger (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.

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.

**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

**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 M_{0,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.

**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.

**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.

**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.

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.

**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.

(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.

(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.) ].

(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.

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 K_{1}=G(R)/E(R)

c) bounded word length in E(R); and

d) normal subgroups of G(R).

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.

**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.

**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.

**Syzygies of binomial ideals and toric Eisenbud-Goto conjecture
(Lev Borisov, April 6, 2011):**

Let p_{1},...,p_{k} 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 p_{i}.
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

**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.

**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.

**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.

**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 *PGL _{n}(C)* that can be
interpreted as cocycles equivalent up to coboundaries. I will start with
a brief introduction to differential Galois theory.

**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.

**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.

**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.

**Reduced K-theory of Azumaya algebras
(Roozbeh Hazrat, December 1, 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.

**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).

**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.

**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).

**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.

**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.

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.

*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.

**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.

f_{11} | f_{12} |

f_{21} | f_{22} |

**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.

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.

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.

**PARTIAL ACTIONS OF GROUPS ON ALGEBRAS
(Miguel Ferrero, June 7, 2005):**

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.

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.

**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):**

**Quasi-Hoph algebras, twisting and the KZ equation
(Edwin Beggs, Dec. 10, 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 *O _{X}*
(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.

**The affine algebra A _{2}^{2} 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 A

**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.

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.

**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):**

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.

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.

The small quantum group

We will consider the smallest modular-invariant subspace in

This is joint work with Eric Friedlander.

To each finite-dimensional module

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.

It is known that the norm map

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.

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

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

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.

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.

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?

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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 @ math.rutgers.edu / January 1, 2018