In this talk we look at a specific problem in probability related to the stochastic versions of the Burgers' and Navier-Stokes equations, and the path taken to construct sufficient assumptions to prove the desired properties, specifically the existence of an invariant distribution. This talk covers material in Stochastic Differential Equations and Stochastic Partial Differential Equations, but also in Real Algebraic Geometry and Perturbation Theory.

I will begin by recounting how I got involved in an industrial collaboration with a company that makes glass tempering furnaces and how younger grad students can seek such opportunities. I will then describe the mathematical model I developed for the company while highlighting challenges that arose due to differences in culture and priorities between academia and industry.

In this talk, I will use my recent project (a fast algorithm for calculating the energy of a many-body quantum system in the random phase approximation) as an outline to present two cool techniques in applied math and show their actual applications in the project. First, we'll see that the trapezoid rule you teach in Calculus, when applied to periodic functions, is far more impressive than you thought. We'll also get a taste of the surprisingly nice properties that come from combining matrix decompositions with randomized algorithms. Finally, as an added bonus, we'll see how Cauchy's integral formula can be used (in this project) to sum N^2 things in O(N) time.

TBA

A very important (and difficult) problem for number theorists is to determine all the rational solutions to a polynomial equation defined over the rational numbers. The oldest nontrivial case, which dates back to Pythagoras, is that of finding all the rational points on a unit circle. In this talk, we will consider the case of elliptic curves, where the rational points have the structure of an Abelian group G under a curiously defined addition law. We will develop some preliminaries and introduce the classical height machinery, a powerful tool which helps us understand the complexity of the points of G. We will look at some important results about the height and about G and see what still remains very elusive.