NTSGrad Spring 2024/Abstracts

This page contains the titles and abstracts for talks scheduled in the Spring 2024 semester. To go back to the main GNTS page for the semester, click here.

1/23

Tejasi Bhatnagar

Stratification in the moduli space of abelian varieties in char p.

This talk will be an introduction to studying moduli space of abelian varieties in characteristic p via different stratifications. This will be a pre-talk for the upcoming Arizona Winter school in March! I'll try and introduce the theory and give an overview of the kinds of questions and objects we'll come across in the winter school.

1/30

Joey Yu Luo

Gross-Zagier formula: motivation

In this talk, I will sketch how to use the modularity theorem to construct lots of rational points in the elliptic curves, based on the idea of Heegner. Among the constructions, we will see how L-functions come into the story, and how the story end up with the Gross-Zagier formula.

2/6

Hyun Jong Kim

A Zoo of L-functions

I will talk about some different kinds of L-functions (and zeta functions) and maybe some problems surrounding them. Here are notes: https://github.com/hyunjongkimmath/GNTS_spring_2024_presentation_notes

2/13

Sun Woo Park

Counting integer solutions of $x^2+y^2=r$ satisfying prime divisibility conditions

As a slight relation to Hyun Jong's talk on zoos of L-functions last week, we'll explore how one can use Dedekind zeta functions over number fields to count integer points lying on a circle of integral radius r centered at the origin and satisfying some prime divisibility conditions. If time allows, we'll see how this counting problem is related to counting isomorphism classes of elliptic curves over Q of bounded naive heights that admit Q-rational 5-isogenies, an application of which is based on joint work with Santiago Arango-Pineros, Changho Han, and Oana Padurariu.

2/20

Eiki Norizuki

Abelian Varieties over C

I will talk about the basics of abelian varieties over the complex numbers, in particular their line bundles, polarization and related topics. Time permitting, I may talk about A_g and the Schottky problem.

3/19

Yifan Wei

Periods and Class Field Theory

We are going to calculate some algebraic integrals on algebraic curves (in particular a twice punctured elliptic curve/Q), figure out their transcendence over Q (to the best of our abilities...). Then we are going to remember class field theory and calculate these integrals cleverly on "the space" (to be revealed in the talk). I'll explain why "the space" is a sane object, and discuss the relation between these integrals and (oh my) Galois representations.

4/16

Hyun Jong Kim

Thesis Defense: Cohen-Lenstra heuristics and vanishing of zeta functions for superelliptic curves over finite fields

Ellenberg-Venkatesh-Westerland proved a Cohen-Lenstra result for imaginary quadratic function fields over finite fields by asymptotically counting points on Hurwitz schemes, which parameterize tamely ramified G-covers of the projective line. Moreover, Ellenberg-Li-Shusterman used the methods of Ellenberg-Venkatesh-Westerland to prove that a fixed complex number vanishes on almost no Zeta functions of hyperelliptic curves over finite fields, with respect to a limit taking the genera of the curves to infinity and then the sizes of the base fields to infinity. I will talk about my thesis work on extending their results to higher degree function fields and curves, including a big monodromy result for the moduli space of superelliptic curves.

4/30

Chenghuang Chen

Bounds for the subsets lacking progressions in {1,2,…,N}

I will talk about progressions in {1,2,…,N}. I will start with bounds for the subsets lacking arithmetic progressions, talk about some history and results in this area. After that, I will focus on bounds for the subsets lacking other progressions like polynomials. If I have more time, I will talk about some details of lacking general polynomials in S. Slijepčević's paper in 2003.