Instructor: Prof. Brad Rodgers
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Registration Deadline: September 19th, 2022
Lecture Times: Monday, Wednesday, and Friday | 12:30 - 1:30 PM (ET)
Office Hours: TBA
Course Dates: September 12th - December 9th, 2022
Mid-Semester Break: October 10th - 14th, 2022
Registration Fee: PSU Students - Free | Other Students - $500 CAD
Prerequisites: Familiarity with probability and analysis and a good background in linear algebra should be sufficient. For part of the course familiarity with the Riemann zeta-function (along the lines of what is covered in a complex analysis course) will be helpful but could likely be picked up fairly quickly even without prior exposure.
Evaluation: Students are expected to attend lectures and work on assignments. There will be four to five assignments given through the term and grades will be based on these assignments.

Capacity Limit: N/A

Format: Online

Course Description

This course will study various properties of large random matrices drawn from the classical compact groups: orthogonal, unitary, or symplectic. Concepts useful for the study of many different random matrix ensembles will be introduced and a portion of the course will emphasize connections between the classical compact groups and problems in number theory. Topics to be covered include distribution of entries, distribution of eigenvalues, correlations and determinantal point processes, symmetric function theory and rudiments of representation theory, central limit theorems for traces of matrices and other linear statistics in eigenvalues, connections to orthogonal polynomials and Toeplitz determinants, moment and ratio formulas, as well as applications of these ideas to the Riemann zeta-function and other L-functions (e.g. the Selberg central limit theorem, moment conjectures, and the work of Montgomery, Hejhal, and Rudnick-Sarnak on correlations of zeros). If there is sufficient time the course will also touch on connections to the gaussian free field and recent work on extremal values of characteristic polynomials or the magnitude of their coefficients.

Resources: Parts of the course will follow the textbook "Random Matrix Theory of the Classical Compact Groups" by E. Meckes (available for free online https://case.edu/artsci/math/esmeckes/Haar_book.pdf). A more detailed schedule of topics will be available in August.