a) random matrices and random point processes, in particular universality in random matrix ensembles, spectral properties of random graphs, general theory of determinantal (pfaffian) random point processes.
b) spectral properties of random and quasi-periodic Schrodinger operators.
c) trace ideals, Fredholm determinants, and their applications.
The suggested research topics are among the most rapidly developing and exciting areas in probability and mathematical physics. In addition we plan to discuss several topics in statistical mechanics, free probability, and Markov chains, such as dynamics of magnetic structures modeled by quantum spin systems, applications of recent developments in free probability (Guionnet, Haagerup, Voiculescu) to ``classical'' random matrix theory, and mixing times of Markov chains. The proposed activities have firm foundations in the areas of strength of the participating faculty members. The involved graduate students will be afforded an excellent opportunity not only to deepen the understanding of their area of thesis research, but also to learn about new problems, methods and results in the adjacent areas of mathematical physics and probability. The proposed research topics allow the active participation of undergraduate students who can explore the subject by analyzing the simplest models or by computer simulation. The activities of the RFG will be centered around a 1 1/2 hour informal weekly reading seminar. One of the main goals of the reading seminar will be to stimulate ongoing research and to formulate the new interesting problems. We will have several visitors who are experts in the proposed areas of research (Alexei Borodin, David Damanik, Ioana Dumitriu, Alice Guionnet, Gunter Stolz, Dmitry Jakobson, Iosif Polterovich, Simone Warzel). We expect that each visitor will give a review talk as well as a seminar talk on their current research.