NTSGrad Fall 2018/Abstracts
This page contains the titles and abstracts for talks scheduled in the Fall 2018 semester. To go back to the main GNTS page, click here.
We will explore topological constraints on the number of rational solutions to a polynomial equation, giving a sketch of Faltings's proof of the Mordell conjecture.
|Asymptotic Equidistribution of Hecke Eigenvalues|
We will talk about Serre's results of the equidistribution of Hecke eigenvalues, wading very slowly through the analysis.
|Growth of class numbers in extensions|
I will explain how class numbers grow in a certain increasing sequence of number fields, why one should expect it based on an analogy with the function field case and the broad context in which this result sits. Time permitting, I will sketch a proof.
|Etale Cohomology: the Streets|
The streets are often dangerous and to survive them one must pick up some basic skills. I will talk about some basic survival skills for the streets of Etale Cohomology.
|Basics of Trace Formula|
This will be a preparatory talk for Thursday's talk. The main goal is to introduce the basic ideas behind the trace formula. Since its statement is mainly formulated in terms of representation theory, I will introduce some notions in representation theory first and explain why number theorists care about it. Then I will give the general statement of trace formula and hopefully do some nontrivial examples. If time allows, I will mention some recent applications of the trace formula in the GGP conjecture, which is a vast generalization of Waldspurger's formula and the Gross-Zagier formula.
|Étale Fundamental Groups - Some Examples|
I will remind everyone what Soumya told us about Étale fundamental groups a few weeks ago and describe some further examples where we can compute some things.