Title: ``Pitman's theorem and the quantum group SL2 in infinite curvature'' Abstract: Pitman's theorem states that a Brownian motion minus twice its current minimum is a Markov process. We will focus on two seemingly distinct approaches to this theorem: Biane's…
Title: "On conjugate systems with respect to completely positive maps" Abstract: In 2010, Dabrowski showed that a von Neumann algebra generated by self-adjoint operators is a factor when they admit a conjugate system. We extend this to the operator-valued case by defining…
Title: "Cover and Hitting Times of Hyperbolic Random Graphs" Abstract: In the hyperbolic random graph model (HRG), nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most at a…
Title: "Fractal geometry methods in semiclassical quantum chaos" Abstract: Quantum Chaos broadly speaking studies the relationship between classical chaos theory and the corresponding quantum dynamics described by the Schrödinger equation. Fractals play a major role in describing the statistics of…
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.…