NTSGrad Fall 2024/Abstracts

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

9/10

Ivan Aidun

Rational Points on Curves, an Introduction to Arithmetic Geometry

Arithmetic geometry is an area of number theory that uses geometry to answer questions about when multivariable polynomials have integer or rational solutions. Already, even the simplest case, finding rational points on curves, offers many interesting facets worth exploring. In this talk I'll introduce several facets of the world of finding points on curves. Although I won't be able to discuss any topic in great depth, I hope to say at least a little bit about: finding points everywhere locally, why are elliptic curves groups, and why does the genus of a curve affect the rational points.

9/17

Amin Idelhaj

Random Walk on Groups

I'll give a random walk through some topics surrounding random walk on finite groups: Fourier analysis, spectral gaps, isoperimetric inequalities, and expander graphs.

9/24

Chenghuang Chen

Exponential Sums in Analytic Number Theory

I will mainly focus on van der Corput's B process for exponential sums in order to fit Thursday's NTS talk. If I have enough time, I will also talk about some related concepts in Montgomery's book "Ten Lectures on the Interface of Analytic Number Theory and Harmonic Analysis".

10/1

Eiki Norizuki

Hodge Numbers of Birational Calabi-Yau Manifolds

This is a prep talk for Thursday's NTS talk. In 1995, Kontsevich introduced motivic integration to prove that Hodge numbers of Birational Calabi-Yau Manifolds are equal. There is an alternative proof using other tools and I will try to outline some of the ingredients of this approach. I may talk about classical Hodge theory, Weil conjectures, p-adic integrations and p-adic Hodge theory.

10/8

Yihan Gu

Surjectivity of l-adic Galois representations

Consider a non-CM elliptic curve over the rationals. Let l be a prime number, we have a Galois group acting on the l-torsion group, which gives us a Galois representation. According to Serre, this representation is surjective for sufficiently large l. On Tuesday, I will introduce an algorithm given by David Zywina which tells us how to find the finite set of primes such that the representation is not surjective for every prime in the set. I will also talk about improved upper bounds of the Serre's result. If we have time, I will briefly introduce another algorithm on Abelian surfaces by Luis V. Dieulefait.

10/15

Jiaqi Hou

Residue symbols and Gauss sums

In this talk, I will introduce the definitions and basic properties of residue symbols and Gauss sums. I will first talk about the evaluations of quadratic Gauss sums and the proof based on the Poisson summation formula used by Dirichlet. I will also discuss cubic residue symbols and cubic Gauss sums, and quartic or n-th power residue symbols if time permits.

10/22

Caroline Nunn

Why is j((1+√-163)/2) a rational number?

The j-invariant is an important invariant of elliptic curves. For an elliptic curve defined over the complex numbers, the j invariant can be calculated using a holomorphic function (in fact, a modular function) called Klein's j function. We will look at the rationality (and more generally the algebraicity) of certain values of the j function and see that this question is closely related to the theory of complex multiplication, that is, the theory of elliptic curves whose endomorphism ring is larger than expected.

10/29

Yifan Wei

Matrix Point Counts on Curves

Classically counting the number of solutions to polynomial equations over a finite field is closely related to global geometric properties of the space defined by those polynomial equations. In this talk we explore the geometry and arithmetic of matrix solutions to polynomial equations over a finite field, possibly from a lot of different directions.

11/5

Ryan Tamura

Orthogonal Shimura Varieties and Kudla's Modularity Conjecture

I will give an introduction to orthogonal Shimura varieties and their associated Kudla cycles. We will start with a slow introduction to modular curves and orthogonal Shimura varieties. In particular, we will discuss Kudla's construction of orthogonal Shimura varieties, and how it naturally yields modular curves. We will then transition towards understanding Kudla divisors, which are Shimura subvarieties constructed from orthogonality conditions. Our main focus will be Borcherds' proof of the Gross-Kohnen-Zagier theorem, which asserts that the associated generating function of Kudla divisors is a weight 3/2 modular form. Time permitted, connections to the Bass and Beilinson-Bloch conjectures which will be discussed.

11/12

Mohammadali Aligholi

Arakelov Geometry

Famous conjectures, such as Mordell and Mordell-Lang, were proved by Faltings and Vojta using "heights" of algebraic varieties, a tool arising from Arakelov geometry, which is often taken as a black box. I'll try to motivate the ideas behind this "geometry" and show you how it unifies arithmetic, algebraic, and complex geometry.

12/3

Ariel Davidovsky

Binary n-ic Forms

Gauss was able to compute the class group of many quadratic rings of integers by studying binary quadratic forms. This is a special case of a more general result that binary n-ic forms parametrize some sort of "rank n" object. We will first explore what binary cubic forms tell us about cubic rings of integers. Time permitting, we will also explore binary quartic forms and binary n-ic forms.

12/10

Tejasi Bhatnagar

Elliptic curves, Hodge structures and the Mumford-Tate conjecture

I’ll talk about complex elliptic curves (and analogously abelian varieties), how they give rise to Hodge structures and their Mumford Tate groups. I’ll compute a few examples and then talk about the Mumford- Tate conjecture, that tells us that the Hodge structure on singular cohomology and the Galois representation coming from \etale cohomology carry the same information.