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
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.
Timetable
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Note: Joint Degree students from Leiden/Delft need to combine information from both the Leiden and Delft MyTimetables to see a complete schedule. This video explains how to do it.
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
As a student, you are responsible for enrolling on time through MyStudyMap.
In this short video, you can see step-by-step how to enrol for courses in MyStudyMap.
Extensive information about the operation of MyStudyMap can be found here.
There are two enrolment periods per year:
Enrolment for the fall opens in July
Enrolment for the spring opens in December
See this page for more information about deadlines and enrolling for courses and exams.
Note:
It is mandatory to enrol for all activities of a course that you are going to follow.
Your enrolment is only complete when you submit your course planning in the ‘Ready for enrolment’ tab by clicking ‘Send’.
Not being enrolled for an exam/resit means that you are not allowed to participate in the exam/resit.
Contact
email: Rajat Hazra
Remarks
none
Software
Starting from the 2024/2025 academic year, the Faculty of Science will use the software distribution platform Academic Software. Through this platform, you can access the software needed for specific courses in your studies. For some software, your laptop must meet certain system requirements, which will be specified with the software. It is important to install the software before the start of the course. More information about the laptop requirements can be found on the student website.