Random Matrix Theory
Code  Completion  Credits  Range  Language 

01TNM  ZK  2  2+0  Czech 
 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, GoldenThompson inequality, Lieb’s theorem, applications of Bernstein’s inequality: sparsification of matrices, matrix multiplication, reconstruction of lowrank 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, noncommutative 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(12), 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:
 Timetable 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 Tue Wed Thu Fri  Timetable for summer semester 2024/2025:
 Timetable is not available yet
 The course is a part of the following study plans:

 Aplikovaná algebra a analýza (compulsory course in the program)
 Aplikované matematickostochastické metody (compulsory course in the program)