December 4, 2018: Patricia Klein

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.

November 27, 2018: James Gossell

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.

November 13, 2018: Hugh Geller

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.

April 17, 2018: Mihran Papikian

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.

April 3, 2018: Kimball Martin

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.

March 13, 2018: Huixi Li

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.

February 27, 2018: Jesse Kass

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.

November 28, 2017: Huixi Li

Fermat’s Theorem on Sums of Two Squares

Huixi Li – Clemson University

Fermat’s theorem on sums of two squares states that an odd prime can be written as the sum of two integer squares if and only if it is congruent to 1 modulo 4. In this presentation I will talk about several proofs of this theorem, and I will compute the ratio of the sum of the bigger terms and the sum of the smaller terms in the representation of such primes as the sum of two squares of positive integers. The result verifies some conjectures of Zhiwei Sun.