# NTSGrad Fall 2018/Abstracts

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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.

## Sept 11

 Brandon Boggess Praise Genus 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.

## Sept 18

 Solly Parenti Asymptotic Equidistribution of Hecke Eigenvalues We will talk about Serre's results of the equidistribution of Hecke eigenvalues, wading very slowly through the analysis.

## Sept 25

 Asvin Gothandaraman Growth of class numbers in $\displaystyle{ \mathbb{Z}_p }$ 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.

## Oct 2

 Soumya Sankar 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.

## Oct 9

 Qiao He 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.

## Oct 16

 Ewan Dalby 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.

## Oct 23

 Niudun Wang Brauer-Siegel Ratio for Abelian varieties over Function Fields I will introduce an analogue of the Brauer-Siegel ratio for Abelian varieties over function fields. Since these are in general complicated, I will compute some examples for elliptic curves.

## Oct 30

 Wanlin Li Siegel Varieties and CM points I will start with introducing modular curves as moduli spaces of elliptic curves with extra structure and the CM points on these curves. Then I will move on to talk about Siegel varieties and CM points for general abelian varieties. This is a prep talk for the Thursday number theory seminar.

## Nov 6

 Sun Woo Park Kloosterman Sums and Kuznetsov Trace Formula This is a preparatory talk for the Thursday number theory seminar talk. I will introduce Kloosterman Sums and state the Kuznetsov trace formula, an equation which relates Kloosterman Sums with the spectral theory of automorphic forms.

## Nov 13

 Sun Woo Park Equidistribution of Heegner Points I will state the definition of Heegner Points and the fundamental theorem of complex multiplication in an adelic formalism. We will also observe that the theorem shows that the Galois orbit of a Heegner point is a torus orbit

## Nov 20

 Solly Parenti What the ad-hell is a modular form? I'll talk about modular forms for awhile. There'll be some fun things and there'll be some automorphic things.

## Nov 27

 Yu Fu Models of Algebraic Varieties over Function Fields of Curves I will introduce some basic notions and background which might be helpful in understanding the upcoming talk on Thursday, where the speaker will talk about the constructions and comparison of Local- Global principles for a variety over function field F, for two choices of local field extension of F. I will recall some basic facts about valuations, valuation rings, rational points of varieties, the definition of models and the geometry of such models, and the relationship between them.

## Dec 4

 Brandon Boggess Height Functions on Varieties Let's count some stuff!

## Dec 11

 Weitong Wang Ax-Schanuel and o-Minimality I'll present the note written by Jacob Tsimerman (with the same title). The goal of this note is to give a geometric interpretation of the Ax- Schanuel theorem, and to give a model-theoretical proof of it.