NTSGrad Fall 2020/Abstracts

This page contains the titles and abstracts for talks scheduled in the Fall 2020 semester. To go back to the main GNTS page, click here.

Sep 15

Qiao He

Local Arithmetic Siegel-Weil Formula at Ramified Prime

In this talk, I will describe a local arithmetic Siegel-Weil formula which relates certain intersection number on U(1,1) Rapoport-Zink space with local density. Via p-adic uniformization, this can be used to establish a global Siegel-Weil formula. The main novelty of this work is that we consider the ramified case. This is a joint work with Yousheng Shi and Tonghai Yang.

Sep 22

Johnny Han

Bounding Numbers Fields up to Discriminant

For those interested in arithmetic statistics, I'll present a quick proof of Schmidt's bound on numbers fields of given degree and bounded discriminant, as well as giving a quick overview of recent improvements on this bound by Ellenberg and Venkatesh.

Sep 29

Brandon Boggess

Dial M_{1,1} for moduli

We'll try to give a brief introduction to moduli problems, with an eye towards moduli of elliptic curves.

Oct 6

Eiki Norizuki

Character Ratio of the Transvection in GL_n(F_q)

This will be a prep talk for Wednesday's NTS talk by Shamgar Gurevich. We will talk about the character ratio of the transvection in GL_n(F_q) and results concerning this quantity.

Oct 13

Di Chen

Recent applications of geometry of numbers.

I will review geometry of numbers and then discuss its applications to bounds of 2-torsion in class groups of number fields (2017) in detail. If time permits, I will also discuss its application to counting number fields with bounded discriminant (2019) briefly.

Oct 20

Yu Fu

Representation stability and the Cohen- Lenstra Conjecture.

I will talk about the tool of representation stability in cohomology and how can one use this to do some algebraic counting over finite field, how it works in the proof of the Cohen- Lenstra conjecture over function field in Jordan's 2015 paper if time permits.

Oct 27

Peter Wei

Belyi’s Theorem and Grothendieck’s dessins d’enfants (children’s drawings)

Belyi’s theorem claims that a complex smooth projective curve X can be defined over a number field, if and only if, there exists a non-constant morphism from X to P^1 as a branched cover with at most three ramification locus. The term dessins d’enfants was coined by Grothendieck in Esquisse d’un programme where he started studying dessins with the knowledge of the “obvious” if part of Belyi’s theorem. I will sketch most of the proof of Belyi’s theorem. If time permits, I will talk more about how Grothendieck was inspired by Belyi’s theorem and the subsequent studies of the faithful action of absolute Galois group Gal(Q^{\bar}/Q) on dessins.

Nov 3

Owen Goff

The Significance of 431: Smoothness, Unexpected Bounds, and the Unsolved Sequence

This talk will cover a seemingly very simple idea that connects to several branches of mathematics -- how do you write numbers as the sum of other numbers that are of the form $2^a3^b$? This talk is somewhat combinatorial in nature and references a few unsolved or recently solved problems.

Nov 10

Ruofan Jiang

Crystals

(This was meant to be a preparation talk for Ziquan Yang happened several weeks ago, but unfortunately I didn’t make it.) I will tell two stories pertaining to crystals. 1. (In char 0) crystals as an alternative way to view algebraic connections. 2.(In char p) resolving the pathology of l-adic cohomology theory when l=p. The intersection of two stories is the theory of crystalline cohomology, which usually contains more arithmetic information than the usual l-adic cohomology theory, and plays an important role in many arithmetic questions. I will focus more on motivating ideas rather than going into details.

Nov 17

Hyun Jong Kim

Line Bundles

I am going to introduce schemes as a generalization of varieties so that I can talk about line bundles. As it turns out, a lot of detail is necessary to even talk about these things, so I am only going to be able to sketch out some details, but not all of them. The central type of detail that I will focus on is "gluing". I will exemplify how gluing works with the projective line and with sections of line bundles on the projective line. I will also briefly illustrate how elements of the class group of a Dedekind domain, an algebraic number theoretic object, are line bundles.

Nov 24

Jiaqi Hou

Reductive Groups

This talk will introduce basic results on the structure of reductive groups over an algebraically closed field.

Dec 1

Tejasi Bhatnagar

Modular forms

This talk will provide an introduction to modular forms. I will not focus much on sketching the details, but rather motivate examples and the important objects we work with. As we go on, we will also encounter Jacobi’s four square problem, Seigel modular forms and the rationality of the Dedekind zeta function.

Dec 8

John Yin

Number Field Sieves

The current fastest algorithm to compute factorizations of numbers is the General Number Field Sieve, which was developed after the Quadratic Number Field Sieve. I will discuss the two algorithms and how they work.

Dec 15

Will Hardt

Some Applications of the Polynomial Method

I will talk about a couple of recent (i.e. 21st century) breakthroughs: Dvir's solution to the Finite Field Kakeya Problem and the Croot-Lev-Pach-Ellenberg-Gijswijt resolution of the cap set problem. The former is the finite field analogue of a harmonic analysis problem in Euclidean space, and the latter is the question of how large subsets of F_q^n can be without containing a 3-term arithmetic progression. Both of these were longstanding open problems whose solutions are short and elegant and centered around basic properties of polynomials.