Title: "Weighted Sums in Free Probability Theory" Abstract: Given free identically distributed self-adjoint random variables $X_1,\dots, X_n$ satisfying certain moment constraints, the free analog of the Berry-Esseen theorem states that the distribution of the normalized sum $S_n = n^{-1/2} \sum_{i=1}^{n}…
Title: "Sparse random matrices, statistical field theory and supersymmetry" Abstract: Solving problems arising from large, sparse random matrices (such as finding their eigenvalue densities) presents substantial challenges for mathematically rigorous techniques revolving around the method of moments and free probability.…
Title: Trees and loop clusters in complete graphs. Abstract: We present a decomposition of the distribution of the subtree connecting several vertices for spanning trees sampled in a complete graph and a short derivation of its asymptotic behavior. Then we…
Title: Norm estimates for epsilon-free semicircles Abstract: Epsilon-freeness, studied by Speicher and Wysoczanski, is a mixture of classical and free independence characterized by a given simple graph. This independence is based on the graph product of groups introduced by Green,…
Title: Operator product expansions and the sine-Gordon model. Abstract: Operator product expansions are a fundamental concept in quantum field theory. Mathematically, there has been progress in understanding them in the setting of conformal field theories (quantum field theories with conformal…
Title: Finite free probability and matrix integrals Abstract: In 2015, Marcus, Spielman, and Srivastava realized that expected characteristic polynomials of sums and products of randomly rotated matrices behave like finite versions of Voiculescu’s free convolution operations. In 2022, I obtained…
Title: Sums of projections with random coefficients. Abstract: I will discuss aspects of spectral theory of infinite sums of the form $\sum_n \kappa_n P_n$, where $P_n$ are rank one projections in a Hilbert space (not necessarily orthogonal to one another),…
Title: Multivariate species sampling models Abstract: Species sampling processes have long provided a fundamental framework for random discrete distributions and exchangeable sequences. However, analyzing data from distinct, yet related, sources, requires a broader notion of probabilistic invariance, with partial exchangeability…
Title: Counting cycles in planar graphs Abstract: Basic Turán theory asks how many edges a graph can have, given certain restrictions such as not having a large clique. A more generalized Turán question asks how many copies of a fixed subgraph…