# Author: Hui Xue

### April 3, 2018: Kimball Martin

## The basis problem

### Kimball Martin – University of Oklahoma

### March 13, 2018: Huixi Li

## Introduction to Modular Forms and Congruence Primes

### Huixi Li – Clemson University

Modular forms are interesting objects in number theory. In this presentation I will first go over the proof of the Lagrange’s four squares theorem using elliptic modular forms. Second I will introduce Hilbert modular forms and Siegel modular forms, with the motivation of generalizing the four squares theorem to totally real fields. Finally I will talk about our recent result on congruence primes for Hilbert Siegel eigenforms. This is joint work with Jim Brown.

### February 27, 2018: Jesse Kass

## How to count lines on a cubic surface arithmetically

### Jesse Kass – University of South Carolina

### November 28, 2017: Huixi Li

## Fermat’s Theorem on Sums of Two Squares

### Huixi Li – Clemson University

Fermat’s theorem on sums of two squares states that an odd prime can be written as the sum of two integer squares if and only if it is congruent to 1 modulo 4. In this presentation I will talk about several proofs of this theorem, and I will compute the ratio of the sum of the bigger terms and the sum of the smaller terms in the representation of such primes as the sum of two squares of positive integers. The result verifies some conjectures of Zhiwei Sun.