Random Matrix Theory

Course guarantor:

Jan Vybíral

Lecturer:

Jan Vybíral

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

Theory of random matrices appeared first in 60's in the 20th century in connection with statistical physics and the theory of nucleis of atoms of heavy metals. The main interest of study is the distribution of eigenvalues of symmetric random matrices. In the 21st century the results of theory of random matrices were applied in theoretical computer science and numerics for design of random algorithms.

Requirements:

Syllabus of lectures:

1. Examples of random matrix ensembles, classes GOE and GUE, Wigner‘s surmise for GOE(2), joint probability density function of spectra of GOE and its proof, Layman‘s classification, Wigner‘s semicircle law

2. Bernstein’s concentration inequality, Golden-Thompson inequality, Lieb’s theorem, applications of Bernstein’s inequality: sparsification of matrices, matrix multiplication, reconstruction of low-rank matrices, randomized matrix decompositions.

Syllabus of tutorials:

Study Objective:

Students will learn classical and modern results and applications from the random matrix theory including Wiegner’s semicircle law, non-commutative concentration inequalities and their applications for construction of randomized algorithms.

Study materials:

M.L. Mehta: Random Matrices 3rd edition, Academic Press, New York (2004)

G. Livan, M. Novaes, P. Vivo: Introduction to Random Matrices: Theory and Practice, Springer, 2018

J. Tropp: An Introduction to Matrix Concentration Inequalities, Foundations and Trends in Machine Learning, 8(1-2), 2015

M. Krbálek and P. Šeba: Statistical properties of the city transport in Cuernavaca (Mexico) and random matrix ensembles, J. Phys. A: Math. Theor. 33 (2000), L229

Note:

Time-table for winter semester 2024/2025:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon	roomTR:205 Vybíral J. 10:00–11:50 (lecture parallel1) Trojanova 13 učebna 205
Tue
Wed
Thu
Fri

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikovaná algebra a analýza (compulsory course in the program)
• Aplikované matematicko-stochastické metody (compulsory course in the program)