FSF3624 HT20 (50776) Random Matrices

Course plan

Random matrix theory has a history going back to mathematical statistics in the 1920's and nuclear physics in the 1950's. In the last 20-30 years the subject has expanded considerably and is now a broad research area. It has connections to many parts of mathematics and other fields, e.g. asymptotic analysis, combinatorics, number theory, statistics, statistical physics, numerical analysis, and communications theory. This means that one can study the area of random matrices from many different points of view and there are many possible approaches to solving problems within the subject. The course will cover some basic aspects of random matrices with an emphasis on basic mathematical concepts and techniques.

 

Lectures

All lecture will be online using zoom. Zoom link for lectures:

Lecture zoom Links to an external site.

After each lecture we will upload lecture notes. 

 

Date Teacher Topic Lecture notes
October 6, at 15.15 Johansson Lecture 1: Introduction to Random Matrices Download LectureNotes1.pdf
October 14, at 15.15 Johansson Lecture 2: Introduction to Discrete Models
October 21, at 15.15 Duits Lecture 3: Determinantal point processes: Discrete case Download F3.pdf
October 28, at 15.15 Duits Lecture 4: Determinantal point processes: general case Download F4.pdf

November 4, at 15.15

 

 Duits Lecture 5: Biorthogonal ensembles Download F5.pdf
November 11, at 15.15 Duits Lecture 6: GUE: Asymptotics  Download F6.pdf
November 18, at 15.15 Duits Lecture 7: Determinantal point processes: Asymptotics
November 25, at 15.15 Johansson Lecture 8: More on discrete models: Tracy-Widom in random growth Download Lecture8.pdf
December 2, at 15.15 Johansson Lecture 9: Extended processes. Dyson Brownian motion.

 

Download Lecture9.pdf
December 9, at 15.15 Schnelli Lecture 10: Semicircle law, Stieltjes transform, Green function. Download Lecture10.pdf
January 20, at 15.15. Schnelli Lecture 11: Non-invariant ensembles Download Lecture11.pdf
January 27, at 15.15 Schnelli Lecture 12: Non-invariant ensemble Download Lecture12.pdf 
February 28 Deadline homework
March 17, at 15.15 Presentations 1
March 24, at 15.15 Presentations 2
March 31 Deadline notes

 

 

Examination

Examination will be based on hand-ins and presentation. The precise form of examination will be determined later and depends on the Covid-19 situation. 

Homework problems: Download RMTExercisesv7.pdf

Note new version Februari 19..

The solutions should be handed in no later than February 28, 2021. For pass you should hand in at least seven of the ten problems.

The other part of the examination will consist of giving a short presentation (max. 20 min) and writing an accompanying note on a topic related to the course. Suggestions for topics ( Download pdf

). Contact for topic distribution: schnelli@kth.se

 

Topic: Student: Presentation date:
Weyl integration formula Nils Hemmingsson March 17
RSK correspondance Petter Restadh March 17
Semicircle law via moment method Aleksa Stankovic March 17
Selberg integral Daniel Ahlsen March 17
Strong Szego theorem Aryaman Jal March 24
Haar unitary matrices Klara Courteaut March 17
Detection thresholds in spiked random matrix models Tianfang Zhang March 17
Weak local semicircle law via Schur complement Joakim Cronvall March 24
Helffer-Sjöstrand formula and rigidity of eigenvalues
Björn Martinsson
 March 24
Kastelyn theory of dimer models
Scott Mason
 March 24
Log-correlated fields and height fluctuations GUE
Wenkui Liu
 March 24
  Philippe Moreillon March 24

Teachers

Maurice Duits, Kurt Johansson and Kevin Schnelli.



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