Basic probability with knowledge in notions of convergence. Knowing basic concepts in functional analysis may be helpful.
Random matrices are matrices whose entries are randomly rolled out. Interestingly, many questions about such matrices, especially about the structure of the eigenvalues and eigenvectors, have a deterministic answer when the size of the matrices approaches infinity. In the past 25 years, random matrices have been studied in more depth in mathematics, and it has become increasingly clear that these objects play an important role at the intersection of quite different mathematical disciplines like operator algebras and probability. We are seeing an increase in the use of random matrices in applied areas such as telecommunications and image processing. In the course we will study the few features of mainly symmetric random matrices. We will keep Wigner matrices as our central focus and introduce the different mathematical techniques which are needed to study the matrices.The topics to be covered are as follows:
1. the empirical measures corresponding to the eigenvalues;
2. method of moments and analytical methods using Stieltjes transform;
3. the behaviour of the largest eigenvalue;
4. large deviation for empirical measures;
5. introduction to free probability theory and mainly use it as a tool box for the combinatorial expressions;
6. ensembles arising out of eigenvalues of adjacency matrix of different random graphs.
The objectives of the course are as follows:
1. Learn two different tools to study empirical measure- one of combinatorial flavour and another analytic.
2. Learn matrix techniques to deal with largest eigenvalue problem which appears in many other context.
3. Introduce the notion of free probability on non-commutative probability spaces so that one can use it in different applications. This will give a glimpse into the link with operator algebras and probability.
4. To make students aware of the interesting subject which is developing very fast.
Mode of instruction
The course will be mainly based on lectures (2 x 45 minutes per class) and students will be provided with the lecture notes. There will be exercises at the end of each week. There will be a small group projects to think and some important papers in this area would be given for some discussion classes.
The final grading -- 40% will be based on homework and 60% will be based on presentations/oral exam on a topic which will be provided.
- An introduction to Random matrices: Greg W. Anderson, Alice Guionnet and Ofer Zeitouni, Cambridge University press 2010.
- Spectral Analysis of Large Dimensional Random Matrices: Zhidong Bai, Jack Silverstein, Springer-Verlag 2010.
- Lectures on the Combinatorics of Free Probability: Alexandru Nica, Roland Speicher, Cambridge University Press 2006.
- Random matrix theory and wireless communication: Antonia Tulino, Sergio Verdú, Found. Trends Comm. Information Theory 1 (2004)
- Topics in random matrix theory: Terence Tao. Graduate studies in mathematics, American Mathematical Society.
- Lectures notes by Manjunath Krishnapur available at http://math.iisc.ernet.in/~manju/RMT/RMT.pdf
- Free probability and random matrices: James A. Mingo and Roland Speicher. Fields Institute Monograph no. 35 (2017)
Interested students can mail me and if more details about the course is needed. There will be two office hours (online or offline) if clarifications are needed. Details to be shared later.