NTSGrad Spring 2020/Abstracts

This page contains the titles and abstracts for talks scheduled in the Spring 2020 semester. To go back to the main GNTS page, click here.

Jan 21

Qiao He

Representation theory and arithmetic geometry

In this talk I will talk about the relation between representation theory and arithmetic geometry. In particular, I will try to discuss several examples that connect representation theory and arithmetic geometry closely. Then if time permits, I will give a brief introduction to trace formula approach, which is the most powerful and promising tools in this field.

Jan 28

Asvin Gothandaraman

Modular forms and class groups

In preparation for Thursday's talk, I will review some concepts from Galois Cohomology. I will also give an introduction to the Herbrand-Ribet theorem.

Feb 4

Johnnie Han

ABC's of Shimura Varieties

I'll present some of the formalization of Shimura varieties, with a strong emphasis on examples so that we can all get a small foothold whenever someone says the term Shimura variety.

Feb 11

Will Hardt and John Yin

Primality Tests Arising From Counting Points on Elliptic Curves Over Finite Fields

We will give some background on counting the rational points on an elliptic curve over a finite field. Then we will apply this theory to a couple of specific elliptic curves and explain how it results in (impractical) primality tests.

Feb 25

Ivan Aidun

Golomb Topologies and Infinitely Many Irreducibles

In 1955, Furstenberg gave a proof of the infinitude of primes by imposing a topology on Z. Under this topology, all open sets are infinite, but if you assume only finitely many primes then {1} is open. A new, similar, topology was introduced by Golomb in 1959, which turned Z^+ into a connected Hausdorff space. A general Golomb topology on a domain R was introduced by Knopfmacher and Porubský in 1997. I will draw on a paper of Clark, Lebowitz-Lockard, and Pollack, and discuss interesting properties of these topologies, and how they relate to properties of the domain R.

Mar 3

Soumya Sankar

Perspectives on Rational Points

This talk is going to be an all-you-can-eat smorgasbord of techniques for finding rational points on curves.

Mar 24

Brandon Boggess

Squares in Arithmetic Progressions

We will see how results about rational points on curves can say something about integers in arithmetic progressions.