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.

## Sept 10

 Brandon Boggess 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.

## Sept 17

 Solly Parenti The Siegel-Weil Formula Theta functions, Eisenstein series, and Adeles, Oh my!

## Sept 24

 Dionel Jamie 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.

## Oct 1

 Qiao He Modularity Theorem 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.

## Oct 7

 Yu Fu Abhyankar's Conjectures 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.

## Oct 15

 Ewan Dalby 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.

## Oct 22

 Will Hardt 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.

## Oct 29

 Soumya Sankar 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.

## Nov 5

 Di Chen Artin-Hecke $\displaystyle{ L }$-functions This talk is an introduction to Artin-Hecke $\displaystyle{ L }$-functions, their basic algebraic and analytic properties, and “equivalence” via class field theory. If time permits, I will briefly introduce what the Rankin-Selberg $\displaystyle{ L }$-function is and what it’s meant for.

## Nov 12

 Hyun Jong Tate's Thesis and Rankin-Selberg theory Yang's talk at the number theory seminar is related to Tate's thesis and Rankin-Selberg theory, so I'm going to introduce both. Tate's thesis generalizes the functional equation of the Riemann zeta function to a function defined using the adeles. Furthermore, the Rankin-Selberg method uncovers a functional equation for an L-function. I will focus on the classical Rankin-Selberg method, but I may also be able to talk about adelic Rankin-Selberg, which seems to be good to know for Yang's talk.

## Nov 19

 Niudun Wang Cohomology Juggle A brief review of Chern class, Etale cohomology and spectral sequences with classical examples and also a slight touch on Steenrod operations which will be a key tool for Tony's talk.