##### Department of Mathematics,

University of California San Diego

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### Math 295 - Mathematics Colloquium

## Professor Van Vu

#### Rutgers University

## Inverse Littlewood-Offord theory, Smooth Analysis and the Circular Law

##### Abstract:

A corner stone of the theory of random matrices is Wigner's semi-circle law, obtained in the 1950s, which asserts that (after a proper normalization) the limiting distribution of the spectra of a random hermitian matrix with iid (upper diagonal) entries follows the semi-circle law. The non-hermitian case is the famous Circular Law Conjecture, which asserts that (after a proper normalization) the limiting distribution of the spectra of a random matrix with iid entries is uniform in the unit circle.\\ Despite several partial results (Ginibre-Mehta, Girko, Bai, Edelman, Gotze-Tykhomirov, Pan-Zhu etc) the conjecture remained open for more than 50 years. In 2008, T. Tao and I confirmed the conjecture in full generality. I am going to give an overview of this proof, which relies on rather surprising connections between various fields: combinatorics, probability and theoretical computer science.

Host: Jozsef Balogh

### April 1, 2010

### 4:00 PM

### AP&M 7321

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