Recorded 26 February 2025. Samuel Johnston of King's College London presents "Horn's problem and free probability" at IPAM's Free Entropy Theory and Random Matrices Workshop. Abstract: In 1962, Horn raised the following problem: Let A and B be n-by-n Hermitian matrices with respective eigenvalues a_1,...,a_n and b_1,...,b_n. What can we say about the possible eigenvalues c_1,...,c_n of A + B? The deterministic perspective is that the set of possible values for c_1,...,c_n are described by a collection of inequalities known as the Horn inequalities. Free probability offers the following alternative perspective on the problem: if (A_n) and (B_n) are independent sequences of n-by-n random matrices with empirical spectra converging to probability measures mu and nu respectively, then the random empirical spectrum of A_n + B_n converges to the free convolution of mu and nu. But how are these two perspectives related? In this talk I will discuss approaches to free probability that bridge between the two perspectives. More broadly, I will discuss how the fundamental operations of free probability (such as free convolution, free compression etc) arise out of statistical physics/optimal transport arguments applied to corresponding finite representation theory objects (hives, Gelfand-Tsetlin patterns, characteristic polynomials, Horn inequalities etc). This talk is based on joint work with Octavio Arizmendi (CIMAT, Mexico), Colin McSWiggen (Academia Sinica, Taiwan) and Joscha Prochno (Passau, Germany). Learn more online at: https://www.ipam.ucla.edu/programs/workshops/free-entropy-theory-and-random-matrices/