## $\alpha$-Siegel-Veech constants for branched cyclic covers of tori

### Martin Schmoll – Clemson University

$\alpha$-Siegel-Veech constants are roughly speaking the quadratic asymptotic constants for weighted counts of the number of geodesics below a given length on flat Riemann surfaces. Here the geodesics length is the parameter with respect to which the asymptotic is taken. Each geodesic loop is weighted by the area of the strip carved out by loops isotopic to the given one, exponentiated with the parameter $\alpha$. This uses the fact, that on the flat surfaces considered closed loops appear in families defining euclidean cylinders. For $\alpha=0$ one has the standard quadratic growth rate of cylinders and for $\alpha=1$ one has the so called area-Siegel-Veech constant related to the sum of Lyapunov exponents of the Kontsevich-Zorich cocycle. The asymptotic constants are normalized, so that to the quadratic growth rates of a unit area flat torus have $\alpha$-Siegel-Veech constants 1.

For (generic) branched cyclic covers of tori with given branching we present a general formula for $\alpha$-Siegel-Veech constants. The formula has arithmetic properties depending only on the degree of the cover and a number we call monodromy factor of the cover. In fact, the formula only depends on a unique decomposition of the covering degree determined by the monodromy factors’ primes and does not depend on any other branching data. Most surprising the area Siegel-Veech constant is 1, if the monodromy factors’ prime factorization contains the same primes as the factorization of the covers degree. This phenomenon was observed through computer experiments and generally conjectured by David Aulicino, who is the coauthor of the research presented.

## The Stiefel-Whitney Characteristic Classes

### Zachary Johnston – Clemson University

The Stiefel-Whitney characteristic classes of a real vector bundle with base space B are a sequence of cohomology classes of B over Z/2Z. Consequently, they are used as an algebraic invariant for distinguishing topological spaces and real vector bundles. More surprisingly, they provide an obstruction to a topological space being the boundary of a smooth compact manifold, given by the cobordism theorem. The Stiefel-Whitney characteristic classes also allow one to conclude some unexpected results classifying which projective spaces can be parallelizable and which can be immersed into a fixed Euclidean space.

## Klingen p^2 vectors for GSp(4)

### Shaoyun Yi – University of Oklahoma

In 2007, Roberts and Schmidt had a satisfactory local theory of new- and oldforms for GSp(4) with trivial central character, in which they considered the vectors fixed by the paramodular groups K(p^n). In this talk, we consider the vectors fixed by the Klingen subgroup of level p^2. We calculate the dimensions of the spaces of these invariant vectors for all irreducible, admissible representations of GSp(4, F), where F is a p-adic field.

## Lech’s inequality and the Stuckrad-Vogel conjecture

### Patricia Klein – University of Kentucky

Abstract: Let $(R, m)$ be a Noetherian local ring, and let $M$ be a finitely generated $R$-module of dimension $d$. Let $e(I,M)$ denote the Hilbert-Samuel multiplicity of $M$ on the ideal $I$. Lech’s inequality states that the set ${\ell(R/I)/e(I,R)}$, as $I$ runs through all $m$-primary ideals, is bounded below by $1/d!e(m,R)$. Stuckrad and Vogel showed that this set is not in general bounded above. However, they conjectured that whenever the completion of $M$ is equidimensional that ${\ell(M/IM)/e(I,M)}$ will indeed be bounded above. We prove this conjecture. This talk is based on joint work with Linquan Ma, Pham Hung Quy, Ilya Smirnov, and Yongwei Yao.

## Gorenstein Injective Modules under Flat Base Change

### James Gossell – Clemson University

Abstract: Let A be a commutative noetherian ring and M an A-module. The injective dimension of M gives a measurement of how badly M fails to be an injective module. In 1975, Hans-Bjorn Foxby introduced a technique to determine the injective dimension of any flat base change of an injective module. This technique can be used to verify that injective modules remain injective under certain flat base changes, such as a localization. In this talk we will discuss this technique and preview future research on a possible extension of Foxby’s result for Gorenstein injective modules.

## DG Algebra Resolutions for Fiber Resolution

### Hugh Geller – Clemson University

Abstract: In 2017, Dr. Saeed Nasseh and Dr. Sean Sather-Wagstaff proved certain rigorous properties of fiber products using the standard homological methods. Since other families of rings have been shown to have similar properties using methods involving commutative differential graded (DG) algebras, there is reason to expect we can achieve the results known by Dr. Nasseh and Dr. Sather-Wagstaff via DG algebras. This talk will address the work needed to complete the first step in applying the DG algebra proof to the case of fiber products.

## Graph laplacians and Drinfeld modular curves

### Mihran Papikian – Penn State University

The relationship between combinatorial laplacians and automorphic forms is an active area of current research with applications to a variety of problems arising in number theory, group theory, and coding theory. I will discuss certain combinatorial laplacians arising in the theory of Drinfeld modular curves, and their applications to estimating congruences between automorphic forms.

## The basis problem

### Kimball Martin – University of Oklahoma

Modular forms arose as a natural object of study in number theory
due to connections with quadratic forms and theta series.  The basis problem, studied in depth by Eichler and others, conversely asks what spaces of modular forms can be generated by theta series, especially theta series attached to
quaternion algebras.  We will discuss a representation-theoretic approach to the basis problem using the Jacquet-Langlands correspondence.  This approach both refines the solution to the basis problem by Hijikata-Pizer-Shemanske for elliptic modular forms and solves the basis problem for Hilbert modular forms.  This has practical applications to computing spaces of modular forms and arithmetic applications to Eisenstein congruences.

## Introduction to Modular Forms and Congruence Primes

### Huixi Li – Clemson University

Modular forms are interesting objects in number theory. In this presentation I will first go over the proof of the Lagrange’s four squares theorem using elliptic modular forms. Second I will introduce Hilbert modular forms and Siegel modular forms, with the motivation of generalizing the four squares theorem to totally real fields. Finally I will talk about our recent result on congruence primes for Hilbert Siegel eigenforms. This is joint work with Jim Brown.

## How to count lines on a cubic surface arithmetically

### Jesse Kass – University of South Carolina

Salmon and Cayley proved the celebrated 19th century result that a smooth cubic surface over the complex numbers contains exactly 27 lines.  By contrast, the count over the real numbers depends on the surface, and these possible counts were classified by Segre.  BenedettiSilhol, FinashinKharlamov and Okonek–Teleman made the striking observation that Segre’s work shows a certain signed count is always 3.  In my talk, I will explain how to extend this result to an arbitrary field.  Although I will not use any homotopy, I will draw motivation from A1-homotopy theory.  This is joint work with Kirsten Wickelgren.