Images courtesy of Wikipedia: a bean machine (left), a realization of a Brownian motion in 3D (middle); and simulated quenching of an Ising model (right)
This is a first-semester course on probability theory at the graduate level, focusing on limit theorems and discrete-time stochastic processes. Prior experience with probability at the undergraduate level is not necessary (though some people may find that it helps intuit the theory we will cover).
This course is useful for any mathematician or scientist who wants to learn probability theory and see how it can be applied to solving models arising from the physical sciences.
After introducing the foundations of probability, we will cover the following topics roughly in this order: independence; 0-1 laws; convergence of random variables and of probability measures; limit theorems; conditional expectations; discrete-time martingales; and Brownian motion.
In the later part of the course, we will apply the theory to study more modern topics of probability. The tentative plan is to cover basics of random matrices (such as the semicircle law for Wigner matrices), random fields, and interacting particle systems (such as the Ising model).
Solid background in (real & complex) analysis and linear algebra at the undergraduate level. Knowledge of measure theory is a plus, though the necessary background will be reviewed in the first week. It goes w/o saying that this is a proof-based course; HW assignments will consist of a mix of proofs and calculations. Please don't hesitate to contact the instructor if you need a permission number to enroll in the course!
We will use S.R.S. Varadhan, "Probability Theory" (Courant Lecture Notes) to cover the standard material. The book can be purchased in print, or downloaded in individual chapters from Prof. Varadhan's web site. Other good references for this material are:
For the material on random matrices, random fields, and interacting particle systems, I will provide handouts and references.