Algebra, Geometry, and Number Theory Seminar

Abstracts

Ayah Almousa - Polarizations of Powers of Graded Maximal Ideals

We give a complete combinatorial characterization of all possible polarizations of powers of the graded maximal ideal (x₁,x₂,…,x_m)ⁿ of a polynomial ring in m variables. We also give a combinatorial description of the Alexander duals of such polarizations. In the three variable case m=3 and also in the power two case n=2 the descriptions are easily visualized and we show that every polarization defines a (shellable) simplicial ball. We conjecture that any polarization of an Artinian monomial ideal defines a simplicial ball. This is joint work with Gunnar Fløystad and Henning Lohne.

The pretalk will be an overview of Stanley-Reisner theory and other useful techniques in combinatorial commutative algebra.

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 F_p points on Jacobians of modular curves.

Tracy Huggins - Finite Groups of Essential Dimension 1

The essential dimension of a group is a nonnegative integer related to the representation theory of the group that measures a group's algebraic complexity. I will show how toric geometry and Cartier duality can be used to classify finite groups of essential dimension 1 over a field of characteristic 0.

Jesse Leo Kass - An introduction to counting curves arithmetically

A long-standing program in algebraic geometry focuses on counting the number of curves in special configuration such as the lines on a cubic surface (27) or the number of conic curves tangent to 5 given conics (3264). While many important counting results have been proven purely in the language of algebraic geometry, a major modern discovery is that curve counts can often be interpreted in terms of algebraic topology and this topological perspective reveals unexpected properties.

One problem in modern curve counting is that classical algebraic topology is only available when working over the real or complex numbers. A successful solution to this problem should produce curve counts over fields like the rational numbers in such a way as to record interesting arithmetic information. My talk will explain how to derive such counts using ideas from A1-homotopy theory. The talk will focus on joint work with Marc Levine, Jake Solomon, and Kirsten Wickelgren including a new result about lines on the cubic surface.

No pretalk.

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 D^b(X) be used as an invariant to answer rationality questions? In particular, what properties of D^b(X) are implied by X being rational, stably rational, or having a rational point? On the other hand, is there a property of D^b(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).

Jiuya Wang - Bounding $\ell$-torsion in class groups of elementary abelian extensions

In this talk, we will introduce a new principle in bounding $\ell$-torsion in class groups. We prove a genuinely unconditional point-wise non-trivial bound on $\ell$-torsion of elementary abelian extensions (non-cyclic). In most cases, it also breaks the GRH bound.

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.

Keller VandeBogert - Trimming Complexes and Applications

In this talk, I will introduce a class of complexes called (iterated) trimming complexes. These complexes can be used to resolve grade 3 ideals defining compressed rings with prescribed socle that is minimal is sufficiently generic cases. In particular, we are able to deduce a structure theory for certain classes of ideals defining compressed rings of type 2. We also explore additional applications of these complexes to resolving subsets of generating sets of an ideal, and produce the graded Betti table for certain subsets of the generating set of the maximal minors of a generic matrix.

The pretalk will introduce complexes/resolutions, some standard homological machinery, and other necessary terminology (with plenty of examples).

Last semester's seminar.