NTSGrad Spring 2023/Abstracts

This page contains the titles and abstracts for talks scheduled in the Fall 2023 semester. To go back to the main GNTS page for the semester, click here.

1/24

Tunus Tuncbilek

The classification of Lie algebras and their representations

I will introduce simple Lie algebras over C and classify them, up to isomorphism, using their Dynkin diagrams and root systems. I will also talk about more advanced results from the literature on Lie algebra representations.

1/31

John Yin

Bilu’s Equidistribution Theorem

Bilu’s Equidistribution Theorem says that any sequence of points with decreasing height is equidistributed on the unit circle. We will show the proof of this and discuss related works if time permits.

2/7

No Talk!

2/14

Hyun Jong Kim

Bounded Height Problem for Dynamically Defined Sets

I will give a survey talk about one of the projects that Laura DeMarco has proposed for the upcoming Arizona Winter School in March. Namely, the problem aims to show that the set of orbit collisions via families of self maps of $\mathbb{P}^1$ defined over $\overline{\mathbb{Q}}$ has bounded height. Notes to this talk are available at https://github.com/hyunjongkimmath/GNTS_spring_2023_presentation_notes

2/21

Yiyu Wang

The Local Monodromy Theorem and the Monodromy-Weight Conjecture

In this talk, I will introduce the celebrated local monodromy theorem in the topology and rephrase it in a number-theoretical way which is due to Grothendieck. I will explain what it means from the topological and Hodge theoretical point of view. This theorem naturally leads to the notion of mondromy filtration, and the Monodromy-Weight conjecture asserts that it coincides with the weight filtration.

2/28

Eiki Norizuki

$o$-minimal structures in number theory

$o$-minimality is an area of model theory that has had applications to many areas including number theory and arithmetic geometry. I will give a gentle introduction to what an $o$-minimal structure is and try to state some results in number theory that can be framed in terms of $o$-minimality. In particular, it can be used to address some problems in point-counting (Pila-Wilkie's theorem) and transcendence number theory (Schanuel's Conjecture). No particular prior knowledge is assumed and it should be accessible to anyone.

3/21

Tejasi Bhatnagar

An introduction to rigid analytic geometry

In this talk, I plan to give a very down-to-earth introduction to geometry in the p-adics. As always, I will mostly motivate new ideas and definitions and mainly talk about how to think about “rigid analytic spaces” with some examples.

3/28

Cancelled

4/4

Sun Woo Park

Selmer groups of twist families of elliptic curves over global function fields

I will try to give an overview of obtaining first moments / probability distribution of Selmer groups of twist families of non-isotrivial elliptic curves over the global function field $\mathbb{F}_q(t)$.

4/11

Hyun Jong Kim

Special Workshop: Machine Learning Tools to Aid Mathematicians

Mathematicians often have to learn new concepts. I will briefly present trouver​, a Python library that I have been developing that uses machine learning models to help this process. In particular, trouver​ can categorize types of mathematical text, identify where notations are introduced in such mathematical text, and attempt to summarize what these notations denote. I will also help those that would like to use trouver get set up with it.

4/18

Yu Fu

Cancelled

4/25

Amin Idehaj

Bruhat-Tits Buildings

Bruhat-Tits buildings are simplicial complexes which arise in the study of reductive groups over p-adic fields. In this talk I will introduce basic notions surrounding buildings, describe those buildings associated with SLn(Qp), and say how this gives rise to Ramanujan graphs in combinatorics.

5/2

Yifan Wei

Coefficients of algebraic power series

When an infinite sum yields an algebraic number, you know you are staring at something quite special. Let's begin to understand such phenomenon starting with power series over C that is ACTUALLY ALGEBRAIC over Q(x). We can't solve high degree polynomials using radicals but who's preventing us from using power series? Join me if you want to see more Riemann surfaces applied to number theory. (And also p-adics if they have time to show up

Special Talk Recording

Will Hardt

On Smyth's Conjecture

In 1986, Smyth asked which tuples of coefficients appear in a linear relation among Galois conjugates over the rationals. Smyth proved a set of necessary conditions and conjectured that they were also sufficient. In this talk, I explain what these conditions are, and present evidence for the conjecture, including heuristic reasoning and a function field analogue. Finally, I briefly mention a connection to slice rank, a notion from additive combinatorics introduced in 2016 by Croot, Lev, Pach, Ellenberg, Gijswijt, and Tao as part of the resolution of the cap set conjecture. This was joint work with John Yin. You must be a member of the seminar to view this recording. For the link to the recording, see https://docs.google.com/document/d/1j7BBA4P6UmDeFgUAClFChLU_4IuQB6Vh-p9n4j8r2FE/edit