The celebrated Alexandrov-Bakelman-Pucci Maximum Principle (often abbreviated as ABP estimate) is a pointwise estimate for solutions of elliptic equations, which was introduced in the 1960s. It was motivated by beautiful geometric ideas and has been a fundamental tool in the study of non-divergent PDEs. More recently, this PDE technique also pays back to geometry - the ABP estimate and its extensions can be used to prove some optimal classical geometric inequalities such as the Isoperimetric and Minkowski inequalites.

How do you choose a random finite abelian group?

A d x d integer matrix M gives a linear map from Z^d to Z^d. The cokernel of M is Z^d/Im(M). If det(M) is nonzero, then the cokernel is a finite abelian group of order det(M) and rank at most d.

What do these groups ‘look like’? How often are they cyclic? What can we say about their p-Sylow subgroups? What happens if instead of looking at all matrices, we only consider symmetric ones? We will discuss distributions on finite abelian p-groups, focusing on ones that come from cokernels of families of random matrices. We will explain how these distributions are related to questions from number theory about ideal class groups, elliptic curves, and sublattices of Z^d.

I will discuss the important phenomenon in mathematics that the solution to a question may reach out to concepts of complexity significantly higher than concepts needed for the formulation of the question.
 

Mathematical and computational modeling have become an indispensible component of research across the sciences. Nevertheless, there are still many examples of research across the sciences where decision making processes are strongly influenced by empirical approaches rather than theory. One of the primary challenges in developing rigorous models of complex processes is capturing the nonlinear interactions of processes across multiple scales in space and time. At the same time, because such models may contain many parameters and can describe wide ranges of behaviors, new methods for parameter estimation and inference are needed as well. In this talk, I will give several examples of new multiscale models and novel applications of parameter inference methodologies in applications ranging from tumor biology to engineering. I will discuss some open problems where there are significant opportunities for future research.

Probability theorey is now being inspired and transformed by challenges of big data. This decade is marked by a fascinating convergence of mathematics, statistics, computer science and electrical engineering. I will describe some data-driven advances in high-dimensional probability and high-dimensional inference.