Special topics proposal: Random matrices
Taught jointly by Nick Ercolani and Hermann Flaschka.
We have made a couple of revisions in the course outline in order to
include some material that should be relevant to the Spring 2005
course on zeta functions proposed by Minhyong Kim. Discussion of
eigenvalue spacings and the zeta function of random graphs will be a
bit more extensive than we had originally intended.
0) General introduction. What are random matrices, what was the
motivation for studying them, why is there so much interest these
days.
1) Gaussian and deformed Gaussian measures on Hermitean matrices.
Applications to the geometry of compact surfaces. A sample: In a
polygon with 2n sides, identify sides in pairs, in all possible
ways. What types of surfaces do you get, and how many? This has
surprising applications to physics.
2) A general framework provided by non-commutative (or "free") random
variables. One finds non-commutative versions of the Gaussian
density, the Central Limit Theorem, and other staples of classical
probability theory.
3) An introduction to eigenvalue spacings. This aspect of random
matrix theory has had great influence on the study of zeros of zeta
functions.
4) Adjacency matrices of random graphs. The Ihara zeta function of a
graph is a close relative of number theory zeta functions. When
things get random, there are almost no rigorous results---just many
conjectures based on computer experiments.
Prerequisite: Real analysis (523 or 527), some undergraduate complex
variables. What little probability theory is used can, and will, be
covered quickly, as needed.
Text: Lecture notes, some published papers, some survey papers.