NTSGrad Spring 2022/Abstracts

This page contains the titles and abstracts for talks scheduled in the Fall 2021 semester. To go back to the main GNTS page, click here.

Sep 14

Hyun Jong Kim

What would Jordan do?

In his notes for students, Jordan has a list of general topics and references in number theory/algebraic geometry/arithmetic geometry that students in arithmetic geometry should be comfortable with after a certain point of time. I will introduce some language used in these general topics for beginners.

Sep 21

Peter YI WEI

The S-Unit equation: p-adic approaches

In this talk, I will go over the history of rational/integral points on curves. In particular, I will introduce a recent proof of the S-unit equation using p-adic period maps, given by Lawrence-Venkatesh.

Sep 28

TBA

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Oct 5

Yifan Wei

Lifting a smooth curve from char p to char 0

Geometry over char p is fascinating or frustrating, depending on who you are. However varieties over char 0 could be enjoyed by geometers of all kinds. We will dicuss one way of lifting a smooth projective variety from char p to char 0. After applying our technique to curves we briefly mention the situation in higher dimensions. And if time permits, we discuss a non-liftable example by Serre.

Oct 12

TBA

TBA

Oct 19

TBA

TBA

Oct 26

Di Chen

Special values of zeta functions at positive even integers

I will introduce Euler's classical result over Q, Klingen-Siegel theorem over totally real number fields, and Zagier's theorems and conjectures over general number fields. I will give many examples and discuss their proofs. If time permits, I will discuss its relation with K-theory.

Nov 2

Jerry Y. Fu

Diophantine approximation: How I learned to stop worrying and love integral points

Diophantine approximation is a crucial tool in studying integral points and Schlickewei's theorem is a very useful theorem in proving finiteness theorems on integral points. In the first part of my talk I will show some elegant proof as applications of the subspace theorem such as Vojta's theorem, the S-unit equation, and then I will introduce main conjectures: Vojta, Mordell, Bombieri and Lang, and their relations to each other.

Nov 9

TBA

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Nov 16

TBA

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Nov 23

Eiki Norizuki

Local Reciprocity

I will talk about local reciprocity, a correspondence of the Galois group of the maximal abelian extension and the multiplicative group. In particular, I will talk about Lubin-Tate theory which constructs this map.

Nov 30

Tejasi Bhatnagar

Counting Number fields: A baby example using Bhargava’s techniques.

In this talk, we will walk through a simple example of counting quadratic extensions using the discriminant. Although, this has been done using classical methods, we will highlight the techniques used by Bhargava through our example, that were essentially used to count the higher degree cases.

Dec 7

Qiao He

Siegel-Weil Formula

Given a positive definite quadratic form X_1^2+...+X_n^2, a natural question to ask is can we find a formula for $r_n(m)=\#\{X\in Z^n| Q(X)=m\}$. Although no explicit formula for $r_n(m)$ is known in general, there do exist an average formula, which is a prototype of the so called Siegel-Weil formula. In this talk, I will introduce Siegel-Weil formula, and show how Deuring's mass formula for supersingular elliptic curve and Hurwitz class number formula follows from Siegel-Weil formula.

Dec 14

John Yin

Heights on Stacks

I will motivate and introduce the definition of a height. Then, I will talk a bit about Arakelov height. This will then lead into a recent paper by Ellenberg, Satriano, and Zureick-Brown, which introduces a notion of height on stacks.