Free probability theory and random matrices

点击翻译按钮获取中文摘要

A few years ago I started a project to understand fluctuations of random matrices with tools from free probability theory. Recently, I have isolated a new concept, called "second order freeness", which is the right underlying structure for many phenomena concerning random matrices and which promises to revolutionize the field. NSERC supported this approach with a Leadership Support Initiative Award and my research in the next years will concentrate on bringing in the harvest. Whereas most of my short term goals lie more on the random matrix side (i.e., understanding random matrices and their fluctuations by using my expertise from operator algebras and free probability theory), I also want to pursue as a long term goal a more speculative project, namely the exciting possibility of a link between fluctuations of random matrices and subfactor theory of von Neumann algebras (the latter resulted in Jones's discovery of new knot invariants). A puzzling link comes from the fact that the work of Jones on planar algebras uses the same kind of combinatorial object as show up in my combinatorial approach to random matrices. I want to investigate whether this is a superficial similarity or whether there is some deeper connection. In the latter case, one can expect a totally new perspective on subfactors, which will change the field dramatically. One of the important features of this project is that it attacks very deep and presumably quite hard problems, but that one can also separate much easier subproblems whose understanding is crucial for achieving the final goal, but which are accessible without a deeper theoretical background, and thus ideal topics for graduate, and even strong undergraduate, students. This will present a unique opportunity to confront young people via accessible concrete problems with one of the most exciting cutting edge research in mathematics. This project will feature the training of 2 undergraduate summer students and 3 graduate students per year.