In this talk, I will develop chronologically the theory of cyclotomic fields, motivated by Fermat's Last Theorem and other interesting problems.

It was known to the ancients that the solutions to the equation $x^2+y^2=1$ can completely described by rational functions in a single parameter $t$: $x=2t/(t^2+1)$ and $y=(t^2-1)/(t^2+1)$. With some more theory, one can show that for equations of degree 3 or higher in two variables, such a parametrization usually doesn't exist. However, generalizations of this problem quickly become difficult: an important open problem in algebraic geometry asks whether the solutions of a cubic equation in 5 variables may be described using 4 parameters. In modern language, is a cubic hyperspace in 5-dimensional affine (or projective) space a rational variety? I will discuss examples of variants of these questions that are tractable, and indicate some areas of current research.

In this talk I will trace a series of classical theorems in Diophantine approximations, culminating in the celebrated Schmidt subspace theorem. Along the way, I will prove some finiteness results on the number of solutions of certain Diophantine equations as well as explore interesting distribution properties of the digits of algebraic numbers. The talk will end with a list of seemingly unrelated problems for which Schmidt subspace theorem has been the main tool for recent advances.

The classification of geometric structures has been used to understand the analytic properties of evolving flows on manifolds. In this talk, I will motivate and explain how one can begin to approach such a classification.

I'll give a summary of ten cool deep learning papers from 2017.

We will describe a surprising phenomenon discovered by Zagier on values of quadratic polynomials. We will then provide an explanation using modular forms.

I will give an introduction to $G_2$ manifolds, setting up the relevant definitions, discussing their properties, and time-permitting outlining some of the classical constructions of these objects.

Given two curves in a plane, how many times do they intersect? How many tangents to a given curve pass through a given point? Given a set of $r$ polynomials in $r$ variables, how many points are fixed? I will introduce some basic algebraic geometry and intersection theory and answer all of these questions. 

The central question of the manifold topology is "classification". Surprisingly, in some sense, this question is harder in dimension 4 compared to higher dimensions. Closely related to the classification question in dimension 4 is the study of knots in 3-manifolds. In this talk, we will discuss slice knots, concordance group and the slice-ribbon conjecture, one of the most important conjectures in low-dimensional topology. We will also discuss the role of invariants to study them.

Special values of the Riemann zeta function and their generalizations arise in many branches of mathematics, including number theory, quantum groups, algebraic geometry, and topology. Starting with an elementary proof due to Calabi that the sum of the reciprocals of the square numbers is $\pi^2/6$, we discuss techniques to study particular values of the Riemann zeta function. We will then study certain generalizations called multiple zeta values and describe some underlying algebraic structure.

It is easy to think of curves defined over $\mathbb{R}$ or $\mathbb{C}$, but we can also define them over finite fields $\mathbb{F}_q$.  Given such a curve, we can consider its extension to finite fields $\mathbb{F}_{q^v}$ and count the number of points.  We will introduce the $\zeta$ function on curves and use it to prove a bound on the number of these points.