Department Colloquium

The problem of finding the smallest eigenvalue of a Hermitian matrix (also called the ground state energy) has wide applications in quantum physics. In this talk, I will first briefly introduce the mathematical setup of quantum algorithms, and discuss how to use textbook quantum algorithms to…

In the last few years there has been an explosion of new results about surfaces in 4-space. In this talk, we will start by discussing various kinds of surfaces and some basic questions about them, like what it means for two of them to be the equivalent. We will then discuss two ways to describe…

Some of the most important problems in combinatorial number theory ask for the size of the largest subset of the integers in an interval lacking points in a fixed arithmetically defined pattern. One example of such a problem is to prove the best possible bounds in Szemer\'edi's theorem on…

When is the distribution of a random variable determined by its moments? For real-valued random variables, this is the content of the classical moment problem. Similar problems exists for random groups. These arose in number theory in the course of understanding the behavior of class groups. In…

The cohomology groups of moduli spaces of curves are important to several mathematical disciplines, from low-dimensional topology and geometric group theory to stable homotopy theory and quantum algebra. Algebraic geometry endows these groups with additional structures, such as Hodge structures…

In this talk, we'll talk about a surprising recent result about the algebraic relations between solutions of a differential equation. We will also give applications to certain recent transcendence and diophantine results.

Abstract

Floer homology theories for 3-manifolds come from many sources Instantons, Seiberg-Witten Monopoles,  Heegaard Floer and Embedded Contact Floer theories.  They have proven to be a powerful tools in low dimensional topology. I’ll try to outline some of their…

Abstract: (Joint with B.Bakker) Periods are integrals of differential forms, and their study spans many branches of mathematics, including diophantine geometry, differential algebra, and algebraic geometry. If one restricts their attention to…

For a finite set X, a family F of subsets of X is said to be increasing if any set A that contains B in F is also in F. The p-biased product measure of F increases as p increases from 0 to 1, and often exhibits a drastic change around a specific value, which is called a "threshold." Thresholds…