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Random Matrices (Zufallsmatrizen)
Dozent: Prof. Dr. Roland Speicher
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- In order to get an idea of the content of the course, please watch the first video of the series.
- If you are interested in this course, please write an email to Roland Speicher (speicher@math.uni-sb.de)
- The time and place of the weekly meetings will be decided during the first week of term.
Course Description
- Students will watch the recorded lectures on Random Matrices (2 videos per week). In our weekly meeting we will discuss this and, in particular, also do the assignments from the lecture notes. (For the published version, see Lectures on Random Matrices by EMS Press.
- By successful participation and passing an oral exam students can acquire 9 credit points.
- The videos are in English, the oral exam can also be taken in German. The language of the weekly meetings will be English, unless all participants speak German.
Content
Random matrices are matrices where the entries are chosen randomly. Surprisingly, it turns out that many
questions on random matrices, in particular on the structure of their eigenvalues, has a deterministic answer
when the size of the matrices tends to infinitiy. During the last few decades random matrix theory has become
a centrepiece of modern mathematics, with relations to many different mathematical fields, as well as
applications in applied subjects like wireless communications, data compression or financial mathematics.
The course will give an introduction into the theory of random matrices and will cover subjects like:
- examples of random matrix ensembles (GUE, Wigner matrices, Wishart matrices)
- combinatorial and analytical methods
- concentration phenomena in high dimensions
- computational methods
- Wigner's semicircle law
- statistics of largest eigenvalue and Tracy-Widom distribution
- determinantal processes
- statistics of longest increasing subsequence
- free probability theory
- universality
- non-hermitian random matrices and circular law
Prerequisites
Prerequisites are the basic courses on Analyis and Linear Algebra. In particular, knowledge on measure and
integration theory on the level of our Analysis 3 classes is assumed.
Background on stochastics is helpful, but not required.
Literature
Publications
- Gernot Akemann, Jinho Baik, Philippe Di Francesco, Oxford Handbooks in Mathematics, 2011,
The Oxford Handbook of Random Matrix Theory - Greg Anderson, Alice Guionnet, Ofer Zeitouni, Cambridge University Press 2010,
An Introduction to Random Matrices - Zhidong Bai, Jack Silverstein, Springer-Verlag 2010,
Spectral Analysis of Large Dimensional Random Matrices - Percy Deift, Courant Lecture Notes 3, Amer. Math. Soc. 1999,
Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach - Percy Deift, Dimitri Gioev, Courant Lecture Notes 18, Amer. Math. Soc. 2009,
Random Matrix Theory: Invariant Ensembles and Universality - Alan Edelman, Raj Rao, Acta Numer. 14 (2005), 233-297,
Random matrix theory - Alan Edelman, Raj Rao, Found. Comput. Math. 8 (2008), 649-702,
The polynomial method for random matrices - Alice Guionnet, Springer-Verlag 2009,
Large Random Matrices: Lectures on Macroscopic Asymptotics - Madan Lal Mehta, Elsevier Academic Press 2004,
Random Matices - James Mingo, Roland Speicher, Springer-Verlag, 2017,
Free Probability and Random Matrices - Alexandru Nica, Roland Speicher, Cambridge University Press 2006,
Lectures on the Combinatorics of Free Probability - Antonia Tulino, Sergio Verdú, Found. Trends Comm. Information Theory 1 (2004), 1-182,
Random matrix theory and wireless communication
Other lectures and lecture notes on random matrices
Postal address
Saarland University
Department of Mathematics
Postfach 15 11 50
66041 Saarbrücken
Germany
Visitors
Saarland University
Campus building E 2 4
66123 Saarbrücken
Germany