David Nguyen (UCSB)

Introduction and Organizational Meeting

Abstract: In 1859 Riemann studied the zeros of the Riemann zeta function and linked them to the distribution of prime numbers. It is fascinating to me how zeros of such purely analytic objects can encode so much arithmetical information about prime numbers. This quarter, our learning seminar, in parallel with the Graduate Number Theory seminar, will investigate arithmetic of zeros of related analytic functions, partly motivated by recent work of B. Rodgers and T. Tao on the Newman conjecture.

Notes.*

Nadir Hajouji (UCSB)

Torsors and Singularities

Abstract: Starting from a smooth, elliptically fibered Calabi-Yau 3-fold \(X\to S\) with generic fiber \(E\), we wish to classify all smooth, genus one fibered Calabi-Yau 3-folds \(Y \to S\) whose generic fiber has a Jacobian isomorphic to \(E\).

In my talk, I will explain what the words in the previous sentence mean, explain some of the motivation and talk about some of the methods I'm using to work on the problem.

Danny Nguyen (UCLA)

Short Generating Functions and their Complexity

Abstract: Short generating functions were first introduced by Barvinok to enumerate integer points in polyhedra. Adding in Boolean operations and projection, they form a whole complexity hierarchy with interesting structure. We study them in the computational complexity point of view. Assuming standard complexity assumption, we show that these functions cannot effectively represent certain truncated theta functions. Along the way, we will draw connection to ordinary number theoretic objects, such as the set of prime or square numbers. This talk assumes no prior knowledge of the subject. Some open questions will be offered at the end. Joint work with Igor Pak.

Notes.

Garo Sarajian (UCSB)

Vinogradov's Mean Value Theorem

Abstract: Exponential sums appear in various analytic settings. Improvements on Vinogradov's Mean Value Theorem and corresponding bounds on exponential sums have led to stronger results in a wide variety of places, from zero-free regions of the Riemann zeta function to Waring's Problem. In this talk, we'll discuss Vinogradov's Mean Value Theorem and some of its applications.

Slides.

Garo Sarajian (UCSB)

A Walk through Waring's Problem

Abstract: A natural generalization of Lagrange's Four Square Theorem, Waring's Problem asks about representing integers as sums of kth powers. After introducing the problem, we will discuss the Hardy-Littlewood Circle Method and some of its variations. We will look into how current results were achieved and how modern tools can be used to further sharpen the bounds to this classical question.

Slides.