Topics in Random Matrix Theory

Random matrices were first introduced in statistics in the 1920’s, but they were made famous by Eugene Wigner’s revolutionary vision. He predicted that spectral lines of heavy nuclei can be modelled by the eigenvalues of random symmetric matrices with
independent entries (Wigner matrices). In particular, he conjectured that the statistics of energy gaps is given by a universal distribution that is independent of the detailed physical parameters. While the proof of this conjecture for realistic physical models is still beyond reach, it has recently been shown that the gap statistics of Wigner matrices is independent of the distribution of the matrix elements. Students will be introduced to the fascinating world of random matrices and presented with some of the basic tools for their mathematical analysis in this course.

Target group: Students with orientation in mathematics, theoretical physics, statistics and computer science.

Prerequisites: No physics background is necessary. Calculus, linear algebra and some basic familiarity with probability theory is expected.

Evaluation: The final grade will be obtained as a combination of the student’s performance on the example sheets and an oral exam.

Teaching format: Lectures

ECTS: 3 Year: 2024

Track segment(s):
Elective

Teacher(s):
László Erdös

Teaching assistant(s):
Volodymyr Riabov