Applied Math Colloquium

Abstract: On the real line, a random walk that can only move in the positive direction is very unlikely to remain close to its origin. After a fixed number of steps, the left tail has a Gaussian profile under minimal assumptions. Remarkably, the same phenomenon occurs when we consider a positive random walk on the cone of positive-semidefinite matrices. After a fixed number of steps, the minimum eigenvalue is described by a Gaussian random matrix model.

This talk introduces a new way to make this intuition rigorous. The methodology addresses an open problem in computational mathematics about sparse random embeddings. The presentation is targeted at a general mathematical audience.

Preprint:  https://arxiv.org/abs/2501.16578

Abstract: The task of simultaneously reconstructing multiple physical coefficients in PDEs from observed data is ubiquitous in applications. We propose an integrated datadriven and model-based iterative reconstruction framework for such joint inversion problems where additional data on the unknown coefficients are supplemented for better reconstructions. Our method couples the supplementary data with the PDE model to make the data-driven modeling process consistent with the model-based reconstruction procedure. This coupling strategy allows us to characterize the impact of learning uncertainty on the joint inversion results for two typical inverse problems. Numerical evidence is provided to demonstrate the feasibility of using data-driven models to improve the joint inversion of multiple coefficients in PDEs.

Abstract: I will discuss the fruitful paradigm of encoding discrete phenomena in complex multivariate polynomials, and understanding them via the interplay of the coefficients, zeros, and function values of these polynomials. For example, the states of a hard-core lattice gas model, the spanning trees or matchings in a graph, and the bases of a matroid can all be viewed in these terms. Over the last decade, this perspective has led to several breakthroughs in computer science, mathematics and statistical physics and an unexpected bridge between distant scientific areas including combinatorics, probability, spectral graph theory, analysis of Markov chains, algebraic geometry, and statistical physics has been built. In this talk, I will discuss several classes of these polynomials and their applications to the traveling salesperson problem, and to counting and sampling problems on matroids.