Graduate Analysis Reading Group

This reading group will work through Barry Simon’s Real Analysis: A Comprehensive Course in Analysis, Part 1, published by the American Mathematical Society. The text provides a modern foundation for graduate-level analysis, integrating functional analysis with classical measure theory. The curriculum is divided into three primary areas: 1. Foundations and Topology: A review of metric spaces and the real number system, followed by a rigorous study of general topological spaces, including compactness, the Stone–Weierstrass theorem, nets, and product topologies. 2. Linear Spaces and Measure Theory: An introduction to Hilbert spaces and Fourier series, followed by an extensive treatment of measure theory. This includes the Riesz–Markov theorem, L^p spaces, Fubini’s theorem, and applications to Gaussian processes and Brownian motion. 3. Functional Analysis and Distributions: An exploration of convexity, Banach spaces, the Baire Category Theorem, and the Hahn–Banach theorem. The final sections cover tempered distributions, the Fourier transform, and an introduction to partial differential equations. Depending on the pace of the group, we may also cover the bonus chapters on probability basics, Hausdorff measure and dimension, and inductive limits. Textbook: Simon, B. (2015). Real Analysis: A Comprehensive Course in Analysis, Part 1. American Mathematical Society. ISBN: 978-1-4704-1099-5.