Probability Theory (MSc course in mathematics)

Roster, Email alias, Lecture notes and assignments: available to those attending the course

Prerequisites: A standard course in probability theory (e.g., at the level of David Williams, Probability with Martingales, Cambridge Univ. Press, 1991), a course in mathematical analysis, some logic and set theory, ability to think independently.

Contents: Foundations of probability theory. Integration and differentiation of measures. Construction of probability measures. Infinite products. Independence. Coupling. The fundamental theorem of probability. Limit theorems. Sums of independent random variables. Martingales. Convergence of probability measures. Elements of ergodic theory. Symmetry and invariance. Introduction to some standard random variables, such as Gaussian random variables, Poisson process, Brownian motion, random trees, Lévy processes. Depending on time, we may discuss some applications in mathematical analysis, in information theory, in queueing theory and in physics.

Goals: Besides those stated in the university catalog, a course in probability should form the basis for other courses in the mathematics department and beyond, ranging from statistics to partial differential equations and modern applications of probability such as, e.g., in finance and in mathematical physics. Probability theory is one of the central topics in mathematics and a foundational course should be in the tool of tricks of everyone. My philosophy in teaching is not to be exhaustive (this is a vacuous statement) but rather explain some central concepts, some of the big ideas in probability, rigorously derive some of its basic theorems, and explain some of its uses.