Date: Tuesday, October 28, 2008
Title: Invertibility of random matrices
Abstract: We will discuss several recent developments related to invertibility
and spectral properties of large random matrices. Consider an $n
\times n$ matrix, whose values are independent identically
distributed random variables. The invertibility questions can be
roughly divided in two categories.
\begin{itemize}
\item Qualitative problems.
\begin{itemize}
\item
What is the probability that a random matrix is invertible?
\end{itemize}
This probability is obviously 1, if the entries are
absolutely continuous. However, in the case of discrete
entries the problem becomes highly nontrivial.
\item Quantitative problems.
\begin{itemize}
\item What is the typical distance between a random matrix and
the set of singular matrices?
\item How is this distance distributed?
\end{itemize}
\end{itemize}
We will briefly survey recent results concerning the qualitative
questions. Most of the talk will be devoted to quantitative results,
which are usually more involved. In particular, we will describe
recent solutions of the von Neumann and SpielmanTeng conjectures
about the behavior of the least singular value. Applications of
these results to classical random matrix theory will also be
considered.
Speaker: Mark Rudelson
Institution: University of Missouri
