(Wednesdays at 2:00 in H525)> RUTGERS ALGEBRA SEMINAR - Spring 2018
(Wednesdays at 2:00 in H425)

The Algebra Seminar meets on Wednesdays, at 2:00-3:00PM in room H425
         (in the Hill Center, on Busch Campus of Rutgers University).
A more comprehensive listing of all Math Department seminars is available.
Here is a link to the algebra seminars in previous semesters


Fall 2018 Seminars (Wednesdays at 2:00 in H525)
5 Sept 12 Dec Patrick Brosnan U.Maryland "Palindromicity and the local invariant cycle theorem"
Spring 2018 Seminars (Wednesdays at 2:00 in H425)
24 Jan Aurélien Sagnier EP-Palaiseau "An arithmetic site of Connes-Consani type for the Gaussian integers"
31 Jan Jozsef Beck   Rutgers   "An annoying question about eigenvalues"
 7 Feb Lev Borisov  Rutgers    "Equations of Cartwright-Steger surface"
21 Feb Dhruv Ranganathan  MIT  "Curves, maps, and singularities in genus one"
28 Feb Rohini Ramadas  Harvard "Algebraic dynamics from topological and holomorphic dynamics"
 7 Mar CANCELLED DUE To snow
15 Mar Nicola Pagani U.Liverpool "The indeterminacy of universal Abel-Jacobi sections" 
21 Mar CANCELLED DUE TO SNOW 
28 Mar Ana-Maria Castravet Northeastern U. "Derived categories of moduli spaces of stable rational curves"
 4 Apr 
11 Apr Chiara Damiolini Rutgers "Conformal blocks attached to twisted groups" 
18 Apr Joe Waldron  Princeton  "Singularities of general fibers in positive characteristic"
 2 May Ben Bakker     U. Georgia  "Hodge theory and o-minimal geometry"
 9 May Antonella Grassi U. Penn  (in H705) "Singularities in  geometry, topology and strings"
Classes end Monday, April 30; Final Exams are May 4-9, 2018
Fall 2017 Seminars (Wednesdays at 2:00 in H525)
13 Sep Louis Rowen  Bar-Ilan Univ "A general algebraic structure theory for tropical mathematics"
20 Sep Nicola Terasca Rutgers   "K-classes of Brill-Noether loci and a determinantal formula"
27 Sep Pham Huu Tiep  Rutgers   "Character levels and character bounds"
4 Oct Dave Jensen     Yale      "Linear Systems on General Curves of Fixed Gonality"
11 Oct Gernot Stroth  Martin-Luther Univ.  "On the Thompson Subgroup"
18 Oct Han-Bom Moon   IAS      "Birational geometry of moduli spaces of parabolic bundles"
 1 Nov Danny Krashen  Rutgers   "Extremely indecomposable division algebras"
 8 Nov Lev Borisov    Rutgers   "Explicit equations of a fake projective plane"
15 Nov Julia Hartmann U. Penn.  "Local-global principles for rational points and zero-cycles"
22 Nov --- no seminar ---      Thanksgiving is Nov. 23; Friday class schedule 
29 Nov Chuck Weibel   Rutgers   "K-theory of line bundles and smooth varieties"
 6 Dec Seth Baldwin  N.Carolina "Equivariant K-theory associated to Kac-Moody groups"
13 Dec Brooke Ullery  Harvard   "Gonality of complete intersection curves"
Classes end December 13; Final Exams are December 15-22, 2017

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


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

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

Here is a link to the algebra seminars in previous semesters

Abstracts of seminar talks


Fall 2018


Palindromicity and the local invariant cycle theorem (Patick Brosnan, December 12, 2018):
In its most basic form, the local invariant cycle theorem of Beilinson, Bernstein and Deligne (BBD) gives a surjection from the cohomology of the special fiber of a proper morphism of smooth varieties to the monodromy invariants of the general fiber. This result, which is one of the last theorems stated in the book by BBD, is a relatively easy consequence of their famous decomposition theorem.
In joint work with Tim Chow on a combinatorial problem, we needed a simple condition ensuring that the above surjection is actually an isomorphism. Our theorem is that this happens if and only if the special fiber has palindromic cohomology. I will explain the proof of this theorem and a generalization proved using the (now known) Kashiwara conjecture. I will also say a little bit about the combinatorial problem (the Shareshian-Wachs conjecture on Hessenberg varieties) which motivated our work.

Spring 2018


Singularities in geometry, topology and strings, Antonella Grassi, May 2, 2018):
I will discuss a "Brieskorn-Grothendieck" program involving certain singularities, Lie algebras and representations. These singularities arise in many different areas of mathematics and physics. I will focus on the case of complex 3 dimensional spaces relating to algebraic geometry, topology and physics. I will disucss local, global and local-to-global properties of threefolds with certain singularities and crieteria for these threefolds to be rational homology manifolds and conditions for threefolds to satisfy rational Poincar\'e duality.
We state a conjecture on the extension of Kodaira's classification of singular fibers on relatively minimal elliptic surfaces to the class of birationally equivalent relatively minimal genus one fibered varieties and we give results in this direction.


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


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


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

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


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


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


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


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


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

Fall 2017


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


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


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


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


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


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


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


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


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

Spring 2017


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


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


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


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


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


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


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

Fall 2016


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


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


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

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


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


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


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


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


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


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


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


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


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

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

Spring 2016


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

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


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


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

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


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


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


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


Symmetrization in tropical algebra (Louis Rowen, Jan. 21, 2016):
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.


Charles Weibel / weibel @ math.rutgers.edu / January 1, 2018