Math 259A

<h1>Math 259A</h1>
<h2>Random matrices and operator algebras.</h2>
<h2>MWF 2-3 in MS7608 (subject to change)</h2>
Dimitri Shlyakhtenko, <a href="mailto:shlyakht@math.ucla.edu">
shlyakht@math.ucla.edu</a> MS7901.  
Office hours: TBA.

<hr>
This will be a course about random matrices, from the perspective of
operator algebras and free probability theory. 
<p>
It turns out that certain tools from von Neumann algebras related to
free probability and ``free analysis'' are very intimately related with
the description of limit behavior of a class of random matrices.
<p>
We will discuss:
<ul>
<li>Basic random matrix theory. Limit eigenvalue density; estimates for
largest eigenvalue. We might possibly talk about more refined eigenvalue
statistics and connections with integrable systems. 

<li>Random matrices ``with a potential''. Related minimization problems
in potential theory. Relations with free probability theory and free
entropy. The approach via free stochastic calculus. Estimates on largest
eigenvalue. Concentration estimates.

<li>Multi-matrix random models and non-commutative probability theory.
Haagerup-Thorbjernsen results on norms of random matrices. Applications.
Biane-Capitaine-Guionnet estimates on free entropy.

<li>Combinatorics of random matrix models. Applications, e.g. using
random matrices to count alternating knots (work of Zinn-Justin and
Zuber).
</ul>
We will assume that the students have basic background in functional
analysis and measure theory. Some knowledge of operator algebras will be
helpful, but not strictly speaking necessary.