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Mathematics Colloquium

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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 non-trivial. \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 Spielman--Teng 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

 

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