A connection is a geometric object that allows to parallel transport vectors along a curve in a domain. A natural question that often arises is whether one can recover a connection inside a domain from the knowledge of the parallel transport along a set of special curves running between boundary points of the domain. In this talk I will discuss this geometric inverse problem in various settings including Riemannian manifolds with boundary and Minkowski space. This problem is related to other inverse problems and is tackled with a range of techniques that I will explore during the talk.
Random matrices now play a role in many areas of theoretical, applied, and computational mathematics. Therefore, it is desirable to have tools for studying random matrices that are flexible, easy to use, and powerful. Over the last fifteen years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that balance these criteria. This talk offers an invitation to the field of matrix concentration inequalities and their applications. This talk is designed for a general audience in mathematics and related fields.
Through the work of Cheeger, Colding, Naber and others we have a deep understanding of the structure of Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature lower bounds. For polarized Kahler manifolds, this was taken further by Donaldson-Sun, who showed that under two-sided Ricci curvature bounds, non-collapsed limit spaces are projective varieties, leading to major progress in Kahler geometry. I will discuss joint work with Gang Liu giving an extension of this result to the case when the Ricci curvature is only bounded from below.
Associativity is ubiquitous in mathematics. Unlike commutativity, its more popular cousin, associativity has for the most part taken a backseat in importance. But over the past few decades, associativity has blossomed and matured, appearing in theories of particle collisions, elliptic curves, and enumerative geometry. We start with a brief look at this history, and then explore the visualization of associativity in the forms of polytopes, manifolds, and complexes. This talk is heavily infused with imagery and concrete examples.