Point Set Topology, Banach and Hilbert Space, Measure Theory, Fourier Series and Transforms, Distribution Theory, Locally Convex Spaces, Basics of Probability Theory, Hausdorff Measure and Dimension.

Selected topics include Bernstein Polynomials, Pointwise Converge of Fourier Series, Lp spaces, Brownian Motion, Measures on Polish Spaces, Haar Measure, Convexity, Alexandroff–Hausdorff and Banach–Mazur theorems, Knaster–Kuratowski Fan, Krein–Milman Theorem, moment problems, fixed point theorems including existence of invariant subspaces for compact operators, Hermite expansions, Nyquist–Shannon sampling theorem, Riesz products, fundamental solutions of classical linear PDEs and the Malgrange–Ehrenpreis theorem, law of large numbers, central limit theorem, law of the iterated logarithm, Poisson processes, Markov chains, Carathéodory construction, inductive limits and ordinary distributions.

Barry Simon is currently an IBM Professor of Mathematics and Theoretical Physics at the California Institute of Technology. He graduated from Princeton University with his Ph.D in Physics. In 2012 Simon won the International Association of Mathematical Physics’ Poincaré Prize for outstanding contributions to mathematical physics. He has authored more than 400 publications on mathematics and physics.