We explore possibility of computing solutions of a certain type of infinitely dimensional Hamilton-Jacobi equations in probability space that arises in the theory of mean field games. Numerical solution to such HJ-PDE was difficult owing to the high dimension of the PDE after discretization of a function space. We propose to utilize a Hopf formula coming from an optimal control approach. The resulting formula is an optimization problem involving a d dimensional HJ-PDE constraint, i.e. the mean field equations, which can be computed using a standard finite difference scheme. In particular, our method will provide us one possible way to compute proximal maps of Wasserstein metrics. They may be of importance in computing optimization problems involving Wasserstein metrics. Our techniques may have applications in optimal transport, mean field games and optimal control in the space of probability densities.

The study of tangent cones in geometric analysis is an important tool in understanding the structure at a singular point of a geometric equation. In this talk I will discuss how to uniquely identify the tangent cone of a Yang-Mills connection with isolated singularity in the complex setting, given an initial assumption on the complex structure of the bundle. I will then discuss applications to a project with the goal of constructing examples of singular G2 instantons, using the twisted connected sum construction. This is joint work with H. Sa Earp and T. Walpuski.

The study of vector bundles on algebraic varieties is a classical topic at the intersection of geometry and commutative algebra. In its algebraic form it is the study of finitely generated projective modules over commutative rings. There are many long-standing conjectures and open questions about algebraic vector bundles, such as: is every topological vector bundle over complex projective space algebraic? In recent years, there have been a number of significant developments in this area made possible using the A^1-homotopy theory of algebraic varieties introduced by Morel and Voevodsky in the late 90s. The talk will provide some background on such questions and discuss some recent joint work with Aravind Asok and Matthias Wendt.

Is grading pain unavoidable?  Probably yes, but we'll give some tips and talk about what we can do make the process better and more fair for everyone.

The invertibility of random matrices with iid entries has been the object of intense study over the past decade, due in part to its role in proving the circular law, as well as its importance in numerical analysis (smoothed analysis). In this talk we review recent progress in our understanding of invertibility for some non-iid models: adjacency matrices of sparse random regular digraphs, and random matrices with inhomogeneous variance profile. We will also discuss estimates for the number of singular values in short intervals. Graph regularity properties play a key role in both problems. Based in part on joint works with Walid Hachem, Jamal Najim, David Renfrew, Anirban Basak and Ofer Zeitouni.