(Wednesdays at 2:00 in H525)

The Algebra Seminar meets on Wednesdays, at 2:00-3:00PM in room H525
         (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 2017 Seminars (Wednesdays at 2:00 in H255)
13 Sep
20 Sep
27 Sep
 4 Oct
11 Oct
18 Oct
25 Oct
13 Dec
Classes end December 13; Final Exams are December 15-22, 2017
Spring 2017 Seminars (Wednesdays at 2:00 in H705)
22 Feb Ryan Shifler  Virginia Tech "Equivariant Quantum Cohomology of the Odd Symplectic Grassmannian"
 1 Mar Chuck Weibel    Rutgers   "The Witt group of surfaces and 3-folds"
 8 Mar Oliver Pechenik Rutgers   "Decompositions of Grothendieck polynomials"
15 Mar no seminar      ------------------- Spring Break ----------
22 Mar Ilya Kapovich UIUC/Hunter College "Dynamics and polynomial invariants for free-by-cyclic groups"
29 Mar Rachel Levanger Rutgers "Interleaved persistence modules and applications of persistent homology to problems in fluid dynamics"
 5 Apr Cristian Lenart Albany-SUNY "Kirillov-Reshetikhin modules and Macdonald polynomials: a survey and applications"
19 Apr Anders Buch     Rutgers     "Puzzles in quantum Schubert calculus"
26 Apr Sjuvon Chung    Rutgers     "Equivariant quantum K-theory of projective space"

Classes end May 1; Final Exams are May 4-10, 2017

Fall 2016 Seminars (Wednesdays at 2:00 in H423)
21 Sept Fei Qi          Rutgers  "What is a meromorphic open string vertex algebra?"
28 Sept Zhuohui Zhang   Rutgers  "Quaternionic Discrete Series"
 5 Oct  Sjuvon Chung    Rutgers  "Euler characteristics in cominuscule quantum K-theory"
12 Oct  Ed Karasiewicz  Rutgers  "Elliptic Curves and Modular Forms"
19 Oct  Natalie Hobson U.Georgia "Quantum Kostka and the rank one problem for sl2m"
26 Oct Oliver Pechenik Rutgers   "K-theoretic Schubert calculus"
 2 Nov Vasily Dolgushev Temple U "The Intricate Maze of Graph Complexes"
 9 Nov Jason McCullough Rider U. "Rees-like Algebras and the Eisenbud-Goto Conjecture"
16 Nov Robert Laugwitz  Rutgers  "Representations of p-DG 2-categories"
23 Nov --- no seminar ---      Thanksgiving is Nov. 24; Friday class schedule 
30 Nov Semeon Artamonov Rutgers  "Double Gerstenhaber algebras of noncommutative poly-vector fields"
 7 Dec Daniel Krashen  U.Georgia "Geometry and the arithmetic of algebraic structures" (Special talk)
14 Dec Angela Gibney  U.Georgia  "Vector bundles of conformal blocks on the moduli space of curves" (Special talk)
Classes end December 14; Final Exams are December 16-23, 2016

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

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

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

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

Here is a link to the algebra seminars in previous semesters

Abstracts of seminar talks

Spring 2017

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

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

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

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

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

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

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

Fall 2016

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Spring 2016

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

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

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

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

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

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

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

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

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

Fall 2015

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

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

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

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

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

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

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

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

Charles Weibel / weibel @ / January 1, 2017