Widely Applied Mathematics Seminar

Tuesday, February 10 | 3 - 4pm

Maxwell Dworkin 334

"Positive Random Walks and Positive-semidefinite Random Matrices"

Speaker: Professor Joel Tropp
Steele Family Professor of Applied & Computational Mathematics, California Institute of Technology

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

Speaker Bio: Joel A. Tropp is Steele Family Professor of Applied & Computational Mathematics at the California Institute of Technology. His research centers on applied mathematics, machine learning, data science, numerical algorithms, and random matrix theory. Some of his best-known contributions include matching pursuit algorithms, randomized SVD algorithms, matrix concentration inequalities, and statistical phase transitions.

Prof. Tropp attained the Ph.D. degree in Computational Applied Mathematics at the University of Texas at Austin in 2004, and he joined Caltech in 2007. He won the PECASE in 2008, and he was recognized as a Highly Cited Researcher in Computer Science each year from 2014–2018. He is co-founder of the SIAM Journal on Mathematics of Data Science (SIMODS), and he was co-chair of the inaugural 2020 SIAM Conference on the Mathematics of Data Science. Prof. Tropp has been elected as a SIAM Fellow, IEEE Fellow, IMS Fellow, and an AMS Fellow. He received the 2025 Richard P. Feynman Prize for Excellence in Teaching at Caltech. He is an invited speaker at the 2026 International Congress of Mathematicians (ICM).