Titles and abstracts

Titles and abstracts will be made available here before the conference.

Martin Weissman:

Title: The arithmetic of arithmetic Coxeter groups

Abstract: In the 1990s, John H. Conway developed a visual approach to the study of integer-valued binary quadratic forms. His creation, the "topograph," sheds light on classical reduction theory, the solution of Pell-type equations, and allows tedious algebraic estimates to be simplified with straightforward geometric arguments. Nowadays, Conway's topograph can be seen to arise from a coincidence between an "arithmetic group" and a "Coxeter group." In this talk, I will survey Conway's approach to integer binary quadratic forms, and show how similar coincidences yield new number theoretic results. These results are joint work with Chris D. Shelley and Suzana Milea.

Background: The talk will be accessible to students familiar with linear algebra (vectors, bases, etc.) and group theory (e.g., the symmetries of polygons and matrix groups like GL(2,R)).

Emily Riehl:

Title: Categorifying cardinal arithmetic.

Abstract: In this talk we’ll prove that a x (b + c) = a x b + a x c via a roundabout method that takes us on a tour through several deep ideas including categorification, the Yoneda lemma, universal properties, and adjunctions.

Sug Woo Shin:

Title: A tale of L-functions

Abstract: Starting from the Riemann zeta function, I will survey the early history on L-functions up to the late 20th century, when Langlands unified different threads of developments

Elena Fuchs:

Title: What network optimization has to do with number theory

Abstract: In the early 70’s, the concept of “expander graphs” was introduced as an optimal model for networks of arbitrary size. Back then, it was not even known whether such graphs exist or not. Today, we not only know that they exist, but have seen their importance come up in a great spectrum of fields in mathematics and computer science. In this talk, we will explore what these graphs are and give an idea about how they have made a splash on the number theory scene.



Postdoc/Grad Student Talks

  • Libby Taylor (Stanford) Combinatorial methods in the arithmetic of curves . I will introduce a method using intersection theory to connect combinatorics to the arithmetic of curves. As an application, we will discuss a result of Katz and Zureick-Brown concerning the number of rational points on a curve over \Q. In particular, we will study the combinatorial methods they use to prove that for a curve X/Q of genus g>1, there is an inequality #X(\Q) \leq #X(\F_q)+2g-2.

  • Kristina Nelson (UC Berkeley) Counting Integer Matrices of Fixed Rank. I will discuss how to use lattices to count (approximately) the number of $n\times m$ integer matrices of fixed rank $k$ that have norm less than $T$. Specifically, I will present a formula of Katznelson for how the number of such matrices grows as $T$ goes to infinity.



Undegraduate Talks:

  • Zhengyuan Shang (Caltech): Construction of Hecke Characters for Three-dimensional CM Abelian Varieties. It is well-known for an elliptic curve with complex multiplication that the existence of a Q-rational model is equivalent to its field of moduli being equal to Q, or its endomorphism ring being the ring of integers of 9 possible fields (*). Murabayashi and Umegaki proved analogous results for abelian surfaces. For three dimensional CM abelian varieties with rational fields of moduli, Chun narrowed down to a list of 37 possible CM fields. In this paper, we show that his list is exact. By constructing certain Hecke characters that satisfy a theorem of Shimura, we prove that precisely 28 isogeny classes of these abelian varieties have Q-models. Therefore the complete analogy to (*) fails here.

  • Ryan Tamura (UC Berkeley): Generalizations of Alder’s conjecture via a conjecture of Kang and Park. Partitions are important combinatorial objects used throughout various fields of mathematics. There are numerous identities regarding partitions, with two famous identities being the Rogers-Ramanujan identities. In this talk I'll provide a brief exposition of partitions and Rogers-Ramunujan type identities. I'll go over a recent conjecture that extended the second Rogers-Ramanujan identity and a proof of this conjecture for all but finitely many cases. Then I'll conclude with a new generalization of this conjecture for arbitrary positive integers. These results are joint work with Adriana L. Duncan, Simran Khunger, and Holly Swisher.

  • Qiyao Yu (Caltech): p-Converse to a Theorem of Gross-Zagier, Kolyvagin, and Rubin for Small Primes. The Birch-Swinnerton-Dyer (BSD) conjecture predicts a relation between arithmetic invariants of an elliptic curve E over a number field K and the behavior of its Hasse-Weil L-function L(E, s) of E at s=1. Consider the case when K is the rationals and p a rational prime. One particular instance of the conjecture claims that the Zp-corank of the p-infinity Selmer group of E being 1 (or 0, respectively) is equivalent to the order of vanishing of the Hasse-Weil L-function being 1 (or 0, respectively). We refer to the direction from the corank to the order of vanishing as the “p-converse.” We study a p-converse theorem for an elliptic curve with complex multiplication, good ordinary reduction at a prime p, and Zp-corank 1. Burungale and Tian have established this theorem in the case of p>3. We identify and resolve some of the difficulties encountered when generalizing this theorem to p=2, 3.

  • Isaac Broudy (UC Berkeley): Seminal Results in the Theory of Integer Partitions and Beyond! In this talk I will introduce some of the major results from theory of integer partitions, with particular focus on the Rogers-Ramanujan identities. The Rogers-Ramanujan identities have led to developments in hypergeometric series, modular forms (specifically mock theta functions), continued fractions, and much more. Finally, we will discuss how the theory of integer partitions have contributed to the development of other areas of mathematics.