NTSGrad Spring 2019/Abstracts
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]\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]GL_n(\mathbb{F}_q)[/math] |
I will discuss the irreducible representations of [math]GL_n(\mathbb{F}_q)[/math]. In particular, I will discuss some ways in which we can understand the structure of representations of [math]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]\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]p[/math]-adic modular forms |
In this talk, I will discuss Serre’s definition of [math]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]\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]\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]p[/math]-Selmer Groups, then focus on the quadratic case of the [math]\ell[/math]-torsion in class groups. |