NTSGrad Fall 2019/Abstracts
This page contains the titles and abstracts for talks scheduled in the Fall 2019 semester. To go back to the main GNTS page, click here.
|Law and Orders in Quadratic Imaginary Fields|
To prepare for Thursday's talk, I will give a gentle introduction to the basic theory of complex multiplication for elliptic curves, beginning with a quick review of elliptic curves over the complex numbers.
|The Siegel-Weil Formula|
Theta functions, Eisenstein series, and Adeles, Oh my!
|On The Discrete Fuglede Conjecture|
It's well known that the set of functions from a finite abelian group to the complex numbers forms an inner product space and that any such function can be written as a unique linear combination of characters which are orthogonal with respect to this inner product. We will look specifically at copies of the integers modulo a fixed prime and I will talk about under what conditions can we take a subset of this group, E, and express a function from E to the complex numbers as a linear combination of characters which are orthogonal with respect to the inner product induced on E.
In this talk, I will describe the modularity theorem of elliptic curve. I will state the main theorem first, and then give some consequences of the theorem. If time permitted, I will try to sketch how to construct Galois representation from modular form and describe Langlands-Tunnell theorem, which will be related with Thursday's talk.
I'll talk about the Etale fundamental group and the equivalence between finite Galois covers and topological monodromy. Then I'll get into the topic of Abhyankar's Conjectures, which deal with Galois covers of curves in positive characteristics.
|Some examples of cohomology in action|
Cohomology plays an important role throughout number theory however it can seem overly abstract and difficult at first, especially if one is not particularly familiar with topology where most people first develop some intuition for it. In this talk I will describe a number of scenarios(some fun, some serious) where we can concretely observe cohomology telling us something which lie outside these normal first examples. No prior knowledge of cohomology will be assumed.
|Spectral Sequences and Completed Cohomology|
In preparation for Thursday's talk, I will begin by introducing spectral sequences, a tool that gives increasingly accurate approximations converging to the homology groups of a filtered chain complex. Next, I will briefly introduce the theory of completed cohomology developed by Calegari and Emerton. I will state a main theorem, which utilizes spectral sequences, then finish with a simple example.
|Chabauty, Coleman and Kim|
This talk is an introduction to the method of Chabauty and Coleman, which is often used to bound the number of rational points on curves of high genus. We will discuss some examples of this method, as well as some limitations. Time permitting, I will talk about non-abelian Chabauty and describe the motivation behind the work of Kim.