## Admission requirements

Measure theory and knowing basic concepts in functional analysis may be helpful.

## Description

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 these 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. free probability

## Course objectives

he 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.

## Timetable

You will find the timetables for all courses and degree programmes of Leiden University in the tool MyTimetable (login). Any teaching activities that you have sucessfully registered for in MyStudyMap will automatically be displayed in MyTimeTable. Any timetables that you add manually, will be saved and automatically displayed the next time you sign in.

MyTimetable allows you to integrate your timetable with your calendar apps such as Outlook, Google Calendar, Apple Calendar and other calendar apps on your smartphone. Any timetable changes will be automatically synced with your calendar. If you wish, you can also receive an email notification of the change. You can turn notifications on in ‘Settings’ (after login).

For more information, watch the video or go the the 'help-page' in MyTimetable. Please note: Joint Degree students Leiden/Delft have to merge their two different timetables into one. This video explains how to do this.

## 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 practise exercises but they wont be compulsory for submission.

## Assessment method

The final grading will be based on a written/oral examination at the end of the course.

## Reading list

- 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)

## Registration

From the academic year 2022-2023 on every student has to register for courses with the new enrollment tool MyStudyMap. There are two registration periods per year: registration for the fall semester opens in July and registration for the spring semester opens in December. Please see this page for more information.

Please note that it is compulsory to both preregister and confirm your participation for every exam and retake. Not being registered for a course means that you are not allowed to participate in the final exam of the course. Confirming your exam participation is possible until ten days before the exam.

Extensive FAQ's on MyStudymap can be found here.

## Contact

email: Rajat Hazra (r.s.hazra[at]math.leidenuniv.nl)