Prof. Dr. Roland Speicher
Dr. Marwa Banna
Ricardo Schnur
Random matrices
(Summer Semester 2018)Lecture Announcement
The default language of the course is English, unless everybody speaks German.
News
Lectures
- Tu 12.00-13.30, lecture hall IV
- Fr 10.15-11.45, seminar room 6
Exam
The final examination will be a written exam and will take place on Tuesday, 07 August, from 10.15 - 13.15 in lecture hall IV. Please be there ahead of time!Excercises
- The exercise sessions take place Mo 8.30-10.00 in seminar room 10 and are held by Alexander Mang.
- Alexander offers a weekly office hour on Tu 16.00-17.00 in room 317.
- Attendance of the exercise sessions is compulsory, if you miss more than 2 sessions please bring a doctor's certificate.
- Exercise sheets will be released on Friday after the lecture and are due the following Friday,
you can either hand them in before the lecture or drop them into letterbox 47. - You can hand in your solutions alone or in pairs.
- You have to obtain at least 50% of the possible points to be eligible to take the final examination.
Assignments
- Assignment 01
- Assignment 02
- Assignment 03
- Assignment 04
- Assignment 05
- Assignment 06
- Assignment 07
- Assignment 08
- Assignment 09
- Assignment 10
- Assignment 11
Course description
Random matrices are matrices where the entries are chosen randomly. Surprisingly, it turns out that manyquestions 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 andintegration 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
- Lecture by Prof. Dr. Folkmar Bornemann, TU München, summer 2011
-
Lecture by Alan Edelman, MIT, spring 2018
- Graduate school on random matrices, Park City Mathematics Institute, summer 2017
- Lecture notes by Todd Kemp, UC San Diego, fall 2013
- Lecture notes by Per-Olof Persson
Updated: 5 September 2018 by Tobias Mai
