Date: Tuesday, October 11, 2011
Title: Random matrices: invertibility, structure, and applications
Abstract: At the heart of random matrix theory lies the realization that the spectrum of a random matrix H stabilizes as the dimensions of H grow to infinity. This phenomenon is captured by the limit laws of random matrix theory, in particular by Wigner's semicircle law, Girko's circular law, and MarchenkoPastur law. These limit laws offer us a clear global and asymptotic picture of the spectrum of H.
In the last few years, a considerable progress was made on the more difficult local and nonasymptotic regimes. In the nonasymptotic regime, the dimensions of H are fixed rather than grow to infinity. In the local regime, one zooms in on a small part of the spectrum of H until one sees individual eigenvalues.
We will see some examples of heuristics, problems and results of the nonasymptotic random matrix theory. We will also discuss lots of examples and applications in various fields  dimension reduction in computer science and signal processing, linear solvers in numerical analysis, spectra of nuclei in physics, and especially the principal component analysis in statistics.
Speaker: Roman Vershynin
Institution: U of Michigan
