(Wednesdays at 2:00 in 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

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" 14 Feb 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 Chiara Damiolini Rutgers "Conformal blocks attached to twisted groups" 15 Mar no seminar ------------------- Spring Break ---------- 21 Mar Patrick Brosnan U.Maryland 28 Mar Ana-Maria Castravet Northeastern U. 4 Apr 11 Apr 18 Apr Joe Waldron Princeton "Singularities of general fibers in positive characteristic" 25 Apr 2 May Ben Bakker U. Georgia TBA 9 May Antonella Grassi U. Penn TBAClasses end Monday, April 30; Final Exams are May 4-9, 2018

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

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"

Here is a link to the algebra seminars in previous semesters

**Conformal blocks attached to twisted groups
(Chiara Damiolini, March 7, 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.

**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.

**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.

**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 Schützenberger (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.

Charles Weibel / weibel @ math.rutgers.edu / January 1, 2017