Topology, Algebraic Geometry, and Dynamics Seminar (TADS)

TADS is a venue for presentation/discussion concerning topology, algebraic geometry, or dynamics (broadly understood).

Attendance is open, and those interested in giving a talk should e-mail title/abstract to any committee member (rgoldin, or alukyane at gmu.edu). Talks are generally 50 minutes. The room is equipped with white boards, and a projector.

Speakers are given the opportunity to have their presentations recorded: YouTube.

Unless noted otherwise, we meet 11:30-12:30 on announced Fridays in 4106 Exploratory Hall. Directions to GMU.

TADS Committee: David Carchedi (chair), Anton Lukyanenko

Spring 2020

February 28, 2020
Akhil Mathew, University of Chicago

A gentle approach to crystalline cohomology
Given a smooth algebraic variety X over C, a classical result of Grothendieck shows that one can recover the singular cohomology of X (with complex coefficients) using only algebraic differential forms. In characteristic p, the analogous construction also works but is less satisfactory: it produces only p-torsion cohomology groups. The de Rham-Witt complex (or crystalline cohomology) provides an integral lift, and yields one of the basic cohomology theories for varieties in characteristic p. In this talk, I will explain a new approach to the de Rham-Witt complex based only on elementary homological algebra. Joint work with Bhargav Bhatt and Jacob Lurie.
February 21, 2020
Owen Gwilliam, UMass Amhrest

Factorization algebras in topology and physics
Factorization homology arises in algebraic topology as a nonlinear generalization of homology theory a la Eilenberg-Steenrod. The first part of my talk will focus on developing the notions of factorization algebra and factorization homology, as articulated by Ayala-Francis and Lurie. In the second part I will discuss how factorization algebras arise naturally in classical and quantum field theory. The goal is to explain recent joint work with Eugene Rabinovich and Brian Williams about field theory on manifolds with boundary, which offers a useful perspective on Kontsevich's deformation quantization of Poisson manifolds as well as the relationship of loop groups to quantum groups via perturbative Chern-Simons theory.
February 7, 2020
Suhyoung Choi, KAIST

Title: Closed affine manifolds with partially hyperbolic linear holonomy
We give some introduction to the field of complete affine n-manifolds, i.e., quotients of the affine n-space by properly discontinuous actions of affine transformation groups. We will try to show that closed manifolds of negative curvature do not admit complete special affine structures whose linear parts are partially hyperbolic in the dynamical sense. We can drop the negative curvature condition. We present our attempt here. (Partially a joint work with Kapovich.)

Fall 2019

December 6, 2019
Grace Work, Girls' Angle and M.I.T.

Title: Discretely shrinking targets in moduli space*
*This talk will begin at 10:30a.m.
Abstract: Consider a nested family of target sets shrinking at a specified rate and a given flow, we want to understand the set of points whose orbits hit these targets infinitely often under the flow. One way to examine this set is to determine under what conditions this set has full measure. This question is closely related to the Borel-Cantelli Lemma and also gives rise to logarithm laws. In joint work with Spencer Dowdall we examine this particular question for Teichmuller flow on the moduli space of unit-area quadratic differentials.
November 22, 2019
Elizabeth Milicevic, Haverford College

The Peterson Isomorphism: Moduli of Curves and Alcove Walks
In this talk, I will explain the combinatorial tool of folded alcove walks, in addition to surveying a wide range of applications in combinatorics, representation theory, and algebraic geometry. As a concrete example, I will describe a labeling of the points of the moduli space of genus zero curves in the complete complex flag variety using the combinatorial machinery of alcove walks. Following Peterson, this geometric labeling partially explains the "quantum equals affine" phenomenon which relates the quantum cohomology of this flag variety to the homology of the affine Grassmannian. This is joint work with Arun Ram.
November 15, 2019
Julianna Tymoczko, Smith College

An introduction to generalized splines
Splines are a fundamental tool in applied mathematics and analysis, classically described as piecewise polynomials on a combinatorial decomposition of a geometric object (a triangulation of a region in the plane, say) that agree up to a specified differentiability on faces of codimension one. Generalized splines extend this idea algebraically and combinatorially: instead of certain classes of geometric objects, we start with an arbitrary combinatorial graph; instead of labeling faces with polynomials, we label vertices with elements of an arbitrary ring; and instead of applying degree and differentiability constraints, we require that the difference between ring elements associated to adjacent vertices is in a fixed ideal labeling the edge. Billera showed that these two characterizations coincide in most cases of real-world interest. In this talk, we describe some of the differences between generalized splines and classical splines, and some of the implications of results about generalized splines for classical splines.
November 1, 2019
Brent Gorbutt, George Mason University

Strings of beads and the equivariant cohomology or Peterson varieties*
*This talk will be at 11:30am
Understanding the product structure of cohomology rings of Grassmann manifolds and flag manifolds is an area of classical interest. For certain special bases of cohomology rings of these spaces, products of basis elements are known to be positive and integral linear combinations of other basis elements for appropriate notions of "positive" and "integral.". In this talk I'll present a positive and integral formula for special basis elements of the equivariant cohomology of the Peterson variety, which is a subvariety of the complete flag manifold. I'll also discuss a previously unknown combinatorial identity that was discovered and used in the proof of the product formula in the equivariant cohomology of the Peterson variety.
October 18, 2019
Changlong Zhong, SUNY Albany

K-theory stable basis of Springer resolutions*
*This talk will be at 10:30am
In this talk I will recall the definition of K-theory stable basis of Springer resolutions, defined by Maulik-Okounkov. It plays important role in calculation of quantum K-theory, and has close relationship with the motivic Chern classes of Schubert varieties from algebraic geometry. I will also mention its relation with affine Hecke algebra action, restriction formula, and wall-crossing matrices. This is joint work with Changjian Su and Gufang Zhao.
October 4, 2019
Anders Buch, Rutgers University

Positivity determines quantum cohomology
I will show that the small quantum cohomology ring of a Grassmannian is, up to rescaling the deformation parameter q, the only graded q-deformation of the singular cohomology ring with non-negative Schubert structure constants. This implies that the (three point, genus zero) Gromov-Witten invariants are uniquely determined by Witten's presentation of the quantum ring and the fact that they are non-negative. A similar statement appears to be true for any flag variety of simply laced Lie type. For the variety of complete flags, this statement is equivalent to Fomin, Gelfand, and Postnikov's conjecture that the quantum Schubert polynomials are uniquely determined by positivity properties. The proof for Grassmannians answers a question of Fulton. This is joint work with Chengxi Wang.
September 20, 2019
Leonardo Mihalcea, Virginia Tech

Cotangent Schubert Calculus
Modern Schubert Calculus studies various intersection rings associated to flag manifolds. All these rings have several common features: they all have a distinguished Schubert basis; the Schubert structure constants count points; the Schubert classes can be defined by equivariant localization. A question with roots in representation theory and microlocal analysis is whether there are good analogues of Schubert classes to study intersection rings of the cotangent bundle of a flag manifold. One answer is given in terms of the characteristic classes of singular subvarieties in the flag manifold, such as the Chern-Schwartz-MacPherson classes. For flag manifolds, these classes are equivalent to the stable envelopes on the cotangent bundle, defined recently by Maulik and Okounkov. I will explain these ideas, and draw parallels with the Schubert Calculus situation. For instance, instead of counting points in three Schubert cycles, in the cotangent situation one takes the Euler characteristic of the intersection of three Schubert cells.
September 6, 2019
Andrei Rapinchuk, University of Virginia

Groups with good reduction
The techniques involving reduction are common in number theory and arithmetic geometry. In particular, elliptic curves and general abelian varieties with good reduction have been the subject of a very intensive investigation over the years. The purpose of this talk is to report on the recent work that focuses on the analysis of good reduction in the context of linear algebraic groups. More precisely, let $G$ be a reductive algebraic group over a field $K$, and assume that $K$ is equipped with a "natural" set $V$ of discrete valuations. We are interested in the $K$-forms of $G$ that have good reduction at all $v$ in $V$. In the case $K$ is the fraction field of a Dedekind domain, a similar question was considered by G.~Harder; the case $K = \mathbb{Q}$ and $V$ is the set of all $p$-adic places was treated in detail by B.H.~Gross and B.~Conrad. I will discuss the emerging results in the higher-dimensional situation where $K$ is the function field $k(C)$ of a smooth geometrically integral curve $C$ over a number field $k$, or even an arbitrary finitely generated field. I will also indicate connections with other questions involving the genus of $G$ (i.e., the set of isomorphism classes of $K$-forms of $G$ having the same isomorphism classes of maximal tori as $G$), the Hasse principle, weakly commensurable Zariski-dense subgroups, etc. (Joint work with V. Chernousov and I. Rapinchuk.)