NTSGrad Spring 2019/Abstracts

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This page contains the titles and abstracts for talks scheduled in the Spring 2019 semester. To go back to the main GNTS page, click here.

Jan 29

Ewan Dalby
Approximating the mean square of the product of the Riemann zeta function with Dirichlet polynomials

Understanding the asymptotics of the mean square of the product of the Riemann zeta function with Dirichlet polynomials allows one to understand the distribution of values of L-functions. I will introduce the problem and describe several results from the paper of Bettin, Chandee and Radziwill who showed how to pass the so called [math]\displaystyle{ \theta=1/2 }[/math] barrier for arbitrary Dirichlet polynomials. This will be a prep talk for Thursdays seminar.

Feb 5

Sun Woo Park
Representations of [math]\displaystyle{ GL_n(\mathbb{F}_q) }[/math]

I will discuss the irreducible representations of [math]\displaystyle{ GL_n(\mathbb{F}_q) }[/math]. In particular, I will discuss some ways in which we can understand the structure of representations of [math]\displaystyle{ GL_n(\mathbb{F}_q) }[/math] , such as parabolic inductions, Hopf algebra structure, and tensor ranks of representations. This is a preparatory talk for the upcoming talk on Thursday.

Feb 12

Hyun Jong Kim
The integrality of the j-invariant on CM points

The j-function, a complex valued function whose inputs are elliptic curves over [math]\displaystyle{ \mathbb{C} }[/math], classifies the isomorphism class of such elliptic curves. We show that, on elliptic curves with complex multiplication (CM), the j-function takes values which are algebraic integers.

Feb 19

Qiao He
L-functions, Heegner Points and Euler Systems

This talk will be about the L-function of an elliptic curve. I will introduce the Gross-Zagier and the Waldspurger formulae, and try to explain why they are deep and useful for the study of L-functions of elliptic curves.

Feb 26

Soumya Sankar
Representation stability and counting points on varieties

In this talk I will describe the Church-Ellenberg-Farb philosophy of counting points on varieties over finite fields. I will talk about some connections between homological stability and asymptotics of point-counts. Time permitting, we will see how this fits into the framework of FI-modules.

Mar 12

Solly Parenti
[math]\displaystyle{ p }[/math]-adic modular forms

In this talk, I will discuss Serre’s definition of [math]\displaystyle{ p }[/math]-adic modular forms. This is a preparatory talk for the Number Theory Seminar on Thursday.

Mar 26

Wanlin Li
The existence of infinitely many supersingular primes for every elliptic curve over [math]\displaystyle{ \mathbb{Q} }[/math]

For the GNTS on visitor's day, I want to present the work of Noam Elkies from his PhD thesis. I will try my best to make this talk completely self-contained, i.e. I will start with defining an elliptic curve and explain what supersingular means.

Apr 2

Weitong Wang
On [math]\displaystyle{ \ell }[/math]-torsion in class groups of number fields

According to Wei-Lun's request, I'll first introduce the big picture of the paper Nonvanishing of Hecke L-Functions and Bloch-Kato [math]\displaystyle{ p }[/math]-Selmer Groups, then focus on the quadratic case of the [math]\displaystyle{ \ell }[/math]-torsion in class groups.

Apr 9

Sang Yup Han
Ergodicity and Sarnak’s Conjecture on Randomness of the Mobius Function

In this talk, I’ll attempt to provide a number theorist’s explanation of flows and ergodicity, using one of our favorite spaces as an example. Then I’ll motivate the subject by presenting Sarnak’s conjecture on the randomness of the Mobius function and it’s corollaries.

Apr 16

Niudun Wang
The march towards Malle's Conjecture

In this talk, I will introduce Malle's conjecture and what is known about it. Then I will present an example using the most complicated group, [math]\displaystyle{ \mathbb{Z}/2\mathbb{Z} }[/math] to show how we count things in practice.

Apr 23

Asvin Gothandaraman
Insufficiency of the Brauer Manin obstruction

I will sketch an example of Poonen showing that the Brauer-Manin obstruction is insufficient in general to detect the non existence of rational points.

Apr 30

Yu Fu
The Cohen-Lenstra Heuristics

I will give a brief introduction to the Cohen-Lenstra Heuristics and will talk about how taking a rank n [math]\displaystyle{ \mathbb{Z}_p }[/math] module with random relations with respect to the Haar measure gives the Cohen- Lenstra distribution on finite abelian groups for class groups of imaginary quadratic fields.

May 7

Brandon Boggess
Stacky Curves and the Generalized Fermat Equation

We will look at the generalized Fermat equation, and see how studying its integral points leads one to study quotient stacks. We will then very quickly turn and run away from the general picture to a particularly simple example of these quotient stacks, the M-curves of Darmon-Granville, and how they can be used to say something about integral points without having to actually know what the hell a stack is.