Modern aspects of random matrix theory : AMS Short Course, Random Matrices, January 6-7, 2013, San Diego, California / Van H. Vu, editor.

Format:

Book

Conference/Event

Author/Creator:

AMS Short Course, Random Matrices, Corporate Author.

Contributor:

Vu, Van, 1970- editor.

Conference Name:

AMS Short Course, Random Matrices (2013 : San Diego, California)

Series:

Proceedings of symposia in applied mathematics ; Volume 72.

Proceedings of Symposia in Applied Mathematics ; Volume 72

Language:

English

Subjects (All):

Random matrices--Congresses.

Random matrices.

Number theory--Congresses.

Number theory.

Physical Description:

1 online resource (176 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2014.

Language Note:

English

Summary:

The theory of random matrices is an amazingly rich topic in mathematics. Random matrices play a fundamental role in various areas such as statistics, mathematical physics, combinatorics, theoretical computer science, number theory and numerical analysis. This volume is based on lectures delivered at the 2013 AMS Short Course on Random Matrices, held January 6-7, 2013 in San Diego, California. Included are surveys by leading researchers in the field, written in introductory style, aiming to provide the reader a quick and intuitive overview of this fascinating and rapidly developing topic. These surveys contain many major recent developments, such as progress on universality conjectures, connections between random matrices and free probability, numerical algebra, combinatorics and high-dimensional geometry, together with several novel methods and a variety of open questions.

Contents:

""Cover""; ""Title page""; ""Contents""; ""Preface""; ""Lecture notes on the circular law""; ""1. Two kinds of spectra""; ""2. Circular law""; ""3. Gaussian case""; ""4. Universal case""; ""5. Comments""; ""References""; ""Free probability and random matrices""; ""Introduction""; ""1. Random matrices and freeness""; ""2. Wigner matrices are asymptotically free""; ""3. The spectrum of non-normal matrices and the Brown measure""; ""References""; ""Random matrix theory, numerical computation and applications""; ""1. A Computational Trick Can Also Be a Theoretical Trick""

""2. Random Matrix Factorization""""3. Stochastic Operators""; ""4. Sturm Sequences and Riccati Diffusion""; ""5. Ghosts and Shadows""; ""6. Universality of the Smallest Singular Value""; ""Acknowledgements""; ""References""; ""Recent developments in non-asymptotic theory of random matrices""; ""1. Introduction""; ""2. Notation and basic definitions""; ""3. Subgaussian random variables""; ""4. Invertibility of a rectangular random matrix""; ""5. Invertibility of a square matrix: absolutely continuous entries""; ""6. Arithmetic structure and the small ball probability""

""7. Putting all ingredients together""""8. Short Khinchin inequality""; ""9. Random unitary and orthogonal perturbations""; ""References""; ""Random matrices: The universality phenomenon for Wigner ensembles""; ""1. Introduction""; ""2. Global and local semi-circular laws""; ""3. Fine-scale spectral statistics: the case of GUE""; ""4. Extending beyond the GUE case I. \index{heat flow} methods""; ""5. Extending beyond the GUE case II. Swapping and the \index{Four Moment Theorem}""; ""6. Sketch of proof of four moment theorem""; ""7. Applications""; ""8. Open questions""; ""References""

""Index""""Back Cover""

Notes:

Description based upon print version of record.

Includes bibliographical references at the end of each chapters and index.

Description based on print version record.

ISBN:

1-4704-1660-3