Math 247A Spring 2011 |
Introduction to Free Probability
Free probability is a very new field of mathematics, invented in the early 1990s. It sits at the intersection of probability theory, functional and complex analysis, and enumerative combinatorics. It has developed both desired and unexpected applications to a surprising range of problems: from random matrix theory to wireless communication; from operator algebras to representation theory of symmetric groups. In free probability, random variables are elements of some (generally non-commutative) algebra, and the classical notion of independence is replaced by one modeled on freeness in group theory. In this general setting, it still makes sense to talk about the distribution of a random variable, and joint distributions of collections of variables; free independence allows joint distributions to be computed from individual ones. The most startling application of this idea is to large random matrices, where it can be used to predict eigenvalues of sums and products of matrices from individual eigenvalue distributions. This course will present an introduction to free probability, concentrating largely on the combinatorial approach. Free independence can be described in terms of the moments (expectations of monomials) of random variables, in a manner that intimately involves the lattice of non-crossing partitions. For a flavour of the ideas involved, see the "WHAT IS..." column from the February 2011 Notices of the AMS: http://www.ams.org/notices/201102/index.html After a brief introduction to the basic tools needed from functional analysis and probability, we will study this lattice and its Moebius function, and develop some of the main theorems of free probability. Depending on audience interest, we may then continue our studies with more in-depth applications and relations to one of the following: eigenvalues or large random matrices; stochastic analysis; von Neumann algebras and free groups; holomorphic functions in the upper-half-plane; representation theory of symmetric groups; parking functions. The textbook for the course will be the excellent LMS Lecture Notes: Lectures on the Combinatorics of Free Probability, by A. Nica and R. Speicher. (This text is quite expensive, though; it will not be required reading, but will be an excellent resource.) No significant background in combinatorics, probability theory, or functional analysis will be needed for this course -- we will develop all ideas from a level appropriate to the audience. Lecture 1 Notes: last update April 5, 2011. Lecture 1. Lecture 2 Notes: last update April 5, 2011. Lecture 2. Lecture 3 Notes: last update April 5, 2011. Lecture 3. Lecture 4 Notes: last update April 5, 2011. Lecture 4. Lecture 5 Notes: last update April 5, 2011. Lecture 5. Lecture 6 Notes: last update April 6, 2011. Lecture 6. Lecture 7 Notes: last update April 9, 2011. Lecture 7. Lecture 8 Notes: last update April 10, 2011. Lecture 8. Lecture 9 Notes: last update April 12, 2011. Lecture 9. Lecture 10 Notes: last update April 19, 2011. Lecture 10. Lecture 11 Notes: last update April 25, 2011. Lecture 11. Lecture 12 Notes: last update April 25, 2011. Lecture 12. Lecture 13 Notes: last update April 26, 2011. Lecture 13. Lecture 14 Notes: last update April 28, 2011. Lecture 14. Lecture 15 Notes: last update May 4, 2011. Lecture 15. Lecture 16 Notes: last update May 3, 2011. Lecture 16. Lecture 17 Notes: last update May 5, 2011. Lecture 17. Lecture 18 Notes: last update May 10, 2011. Lecture 18. Lecture 19 Notes: last update May 12, 2011. Lecture 19. Lecture 20 Notes: last update May 14, 2011. Lecture 20. Lecture 21 Notes: last update May 17, 2011. Lecture 21. Lecture 22 Notes: last update May 19, 2011. Lecture 22. Lecture 23 Notes: last update May 23, 2011. Lecture 23. Lecture 24 Notes: last update May 25, 2011. Lecture 24. Lecture 25 Notes: last update May 26, 2011. Lecture 25. Lecture 26 Notes: last update May 30, 2011. Lecture 26. Lecture 27 Notes: last update June 2, 2011. Lecture 27. |