Horn's Problem and Free Probability

This lecture explores the fascinating intersection of Horn's problem and free probability theory, presented by Samuel Johnston from King's College London at IPAM's Free Entropy Theory and Random Matrices Workshop. Delve into Horn's 1962 problem concerning eigenvalues of Hermitian matrices: given matrices A and B with known eigenvalues, discover what can be determined about the eigenvalues of their sum. Examine two perspectives on this problem—the deterministic approach using Horn inequalities and the free probability approach involving random matrices with converging empirical spectra. Learn how these seemingly different viewpoints connect through statistical physics and optimal transport arguments. The presentation bridges fundamental operations of free probability (free convolution, free compression) with finite representation theory objects (hives, Gelfand-Tsetlin patterns, characteristic polynomials). This research represents collaborative work with Octavio Arizmendi, Colin McSWiggen, and Joscha Prochno.