Kavita Ramanan - Virginia Mathematics Lectures - November 13-15, 2023
Event start date: Monday, November 13, 2023
Virginia Mathematics Lectures »
Kavita Ramanan (Brown University) - Tales of high-dimensional distributions
Though they have a common thread, the three lectures will be self-contained, and can mostly be followed independently of each other.
- Lecture 1: November 13 | 5-6pm | Wilson 402
- Lecture 2: November 14 | 5-6pm | Nau 101
- Lecture 3: November 15 | 5-6pm | Wilson 402
Lecture 1: Tails of random projections: where probability meets geometry
Date: November 13 | 5-6pm
Location: Wilson 402
Abstract: In several areas of mathematics, including probability theory, statistics and asymptotic convex geometry, one is interested in high-dimensional objects, such as measures, data or convex bodies. One common theme is to try to understand what lower-dimensional projections can say about the corresponding high-dimensional objects. We show how this line of inquiry leads to geometric generalizations of several classical results in probability, and demonstrate how the tail behavior of lower-dimensional projections can be used to glean insight into high-dimensional measures.
Lecture 2: A probabilistic approach to the geometry of p-Schatten balls
Date: November 14 | 5-6pm
Location: Nau 101
Abstract: The lp spaces are a canonical example of Banach spaces whose geometry is well understood and has been fruitfully studied using probabilistic methods. In this talk, we describe how a completely different set of probabilistic tools can be used to probe the geometry of their less well understood non-commutative analogs, the p-Schatten spaces of matrices, and touch upon how these results are linked with some open questions in convex geometry. Along the way we describe some new results on Haar measure on the orthogonal group (or more generally, Stiefel manifolds) that may be of independent interest.
Lecture 3: A unimodular Sanov's theorem and interacting particle systems on sparse random graphs
Date: November 15 | 5-6pm
Location: Wilson 402
Abstract: Interacting particle systems consist of collections of stochastically evolving particles indexed by the vertices of a graph, where each particle’s state depends directly only on the states of neighboring vertices in the graph. Such systems model a wide range of physical phenomena including magnetism, the spread of diseases and information, neuronal spiking and opinion dynamics. An important goal is the characterization of both typical and atypical (or large deviations) behaviour of macroscopic quantities, captured by empirical measures, of these systems as the number of vertices goes to infinity. While classical work has mostly focused on the case when the underlying graph is dense, where mean-field theory is applicable, most real-world networks are sparse. We describe recent developments in the sparse setting, including a generalization of the classical Sanov theorem from large deviations theory to the setting of marked unimodular random graphs.
Last updated: Saturday, September 23, 2023