2019-2020 Academic Year


Date Room Speaker Title Host
Sep 6
2:30pm
LC 303B Alicia Lamarche
University of South Carolina
Derived Categories, Arithmetic, and Rationality Questions (local)
Sep 20
2:30pm
LC 303B Robert Vandermolen
University of South Carolina
Curious Kernels in Geometric Invariant Theory (local)

Palmetto Number Theory Series
September 21-22 at UNC Charlotte

Sep 27
2:30pm
LC 303B Lea Beneish
Emory University
Module constructions for certain subgroups of M24 Thorne
Sep 27
3:30pm
LC 303B Jackson Morrow
Emory University
Non-Archimedean entire curves in closed subvarieties of semi-abelian varieties Thorne

Fall Break

Oct 25
2:30pm
LC 303B Eric Sharpe
Virginia Tech
A proposal for nonabelian mirrors in two-dimensional theories Ballard

Modular Forms, Arithmetic, and Women in Mathematics
November 1-3 at Emory

Nov 8
2:30pm
LC 303B Sabrina Pauli
University of Oslo
Lines on a Quintic Threefold Kass
Nov 15
2:30pm
LC 303B Jiuya Wang
Duke University
(date to be confirmed) Thorne
Nov 22
2:30pm
LC 303B Tracy Huggins
University of South Carolina
TBA (local)

Thanksgiving Break

Dec 6
2:30pm
LC 303B Candace Bethea
University of South Carolina
TBA (local)

Palmetto Number Theory Series
December 7-8 at Clemson


Winter Break

Jan 17
2:30pm
LC 303B -
- -
Jan 24
2:30pm
LC 303B -
- -
Jan 31
2:30pm
LC 303B -
- -
Feb 7
2:30pm
LC 303B -
- -
Feb 14
2:30pm
LC 303B -
- -
Feb 21
2:30pm
LC 303B -
- -
Feb 28
2:30pm
LC 303B -
- -

Arizona Winter School on Nonabelian Chabauty
March 7-11 in Tucson, AZ (funding deadline: November 8)

Mar 6
2:30pm
LC 303B -
- -

Spring Break

Mar 20
2:30pm
LC 303B -
- -
Mar 27
2:30pm
LC 303B -
- -
Apr 3
2:30pm
LC 303B -
- -
Apr 10
2:30pm
LC 303B -
- -
Apr 17
2:30pm
LC 303B -
- -
Apr 24
2:30pm
LC 303B -
- -


Abstracts

Lea Beneish - Module constructions for certain subgroups of M24

For certain subgroups of M24, we give vertex operator algebraic module constructions whose associated trace functions are meromorphic Jacobi forms. These meromorphic Jacobi forms are canonically associated to the mock modular forms of Mathieu moonshine. We describe two kinds of vertex operator algebra construction in this work. Both are related to the Conway moonshine module, and the second employs a technique introduced by Anagiannis-Cheng-Harrison. Using the second construction we are able to give an explicit realization of trace functions whose integralities are equivalent divisibility conditions on the number of Fp points on Jacobians of modular curves.

Alicia Lamarche - Derived Categories, Arithmetic, and Rationality Questions

When trying to apply the machinery of derived categories in an arithmetic setting, a natural question is the following: for a smooth projective variety X, to what extent can Db(X) be used as an invariant to answer rationality questions? In particular, what properties of Db(X) are implied by X being rational, stably rational, or having a rational point? On the other hand, is there a property of Db(X) that implies that X is rational, stably rational, or has a rational point?

In this talk, we will examine a family of arithmetic toric varieties for which a member is rational if and only if its bounded derived category of coherent sheaves admits a full étale exceptional collection. Additionally, we will discuss the behavior of the derived category under twisting by a torsor, which is joint work with Matthew Ballard, Alexander Duncan, and Patrick McFaddin.

Jackson Morrow - Non-Archimedean entire curves in closed subvarieties of semi-abelian varieties

The conjectures of Green-Griffiths-Lang-Vojta predict the precise interplay between different notions of hyperbolicity: Brody hyperbolic, arithmetically hyperbolic, Kobayashi hyperbolic, algebraically hyperbolic, and groupless. In his thesis (1993), W. Cherry defined a notion of non-Archimedean hyperbolicity; however, his definition does not seem to be the "correct" version, as it does not mirror complex hyperbolicity. In recent work, A. Javanpeykar and A. Vezzani introduced a new non-Archimedean notion of hyperbolicity, which fixed this issue and also stated a non-Archimedean version of the Green-Griffiths-Lang-Vojta conjecture.

In this talk, I will discuss complex and non-Archimedean notions of hyperbolicity and a proof of the non-Archimedean Green-Griffiths-Lang-Vojta conjecture for closed subvarieties of semi-abelian varieties.

Sabrina Pauli - Lines on a Quintic Threefold

A classical result says that there are 27 complex lines on a smooth cubic surface. There are two types of real lines on cubic surfaces called hyperbolic and elliptic, and the number of real hyperbolic lines minus the number of real elliptic lines on a real smooth cubic surface is equal to 3. There is a new approach to solve questions in enumerative geometry inspired by Morel’s degree map in A1-homotopy theory which refines the Brouwer degree. Morel’s degree map takes values in the Grothendieck-Witt group GW(k). Kass and Wickelgren define the type of a line on a smooth cubic surface defined over a field k as an element in GW(k) and show that the sum over the types of the lines on a cubic surface equals 15 < 1 > +12 < −1 >∈ GW(k) which recovers both the complex (take the rank) and real count (take the signature)

Finashin and Kharlamov define the type of a real line on a general degree 2n−1-hypersurface in Pn. In my talk I will explain how their definition can be generalized to the definition of the type of a line defined over an arbitrary field k and show that the sum over the types of the lines on a general quintic threefold is equal to 1445 < 1 > +1430 < −1 >∈ GW(k).

Eric Sharpe - A proposal for nonabelian mirrors in two-dimensional theories

In this talk we will describe a proposal for nonabelian mirrors to two-dimensional (2,2) supersymmetric gauge theories, generalizing the Hori-Vafa construction for abelian gauge theories. Specifically, we will describe a construction of B-twisted Landau-Ginzburg orbifolds whose classical physics encodes Coulomb branch relations (quantum cohomology), excluded loci, and correlation functions of A-twisted gauge theories. The proposal has been checked in a wide variety of cases, but the talk will focus on exploring the proposal in two examples: Grassmannians (constructed as U(k) gauge theories with fundamental matter), and SO(2k) gauge theories.

Robert Vandermolen - Curious Kernels in Geometric Invariant Theory

The study of Derived Categories, first introduced by Grothendieck and his student Verdier, has come a long a way in describing deep connections between algebraic geometry, commutative algebra, representation theory, number theory, symplectic geometry, and theoretical physics. Despite that a lot of work has been done with derived categories there are still many open questions and interesting conjectures. One example is from the mid 90's and early 2000's where Bondal, Orlov and (independently) Kawamata conjectured that if two smooth complex varieties are related by a flop then they should be derived equivalent. It has been shown by Reid and others that flops can be equivalently explained by wall crossings from variations of geometric invariant theory. Using this approach Ballard Diemer, Favero have suggested a Kernel as a candidate to realize this conjectured derived equivalence. In recent joint work with Ballard, Chidambaram, Favero, and McFaddin, we have shown that a generalization of this kernel realizes the derived equivalence of the Grassmann Flop, which was first introduced by Donovan and Segal. In this talk we will examine some of the curious algebraic and geometric properties of a brand new generalization which describes these previously studied kernels and may describe many more wall crossings.



Last year's seminar.