R A N D O M    M A T R I C E S     A N D     N U M B E R     T H E O R Y ,    F a l l  2 0 1 2 

Lectures: Tuesday and Thursday, 11.30-1pm, in Science Center 101b.

Lecturer: Paul Bourgade, office hours Tuesday 5-6pm, you also can email me (bourgade@math.harvard.edu) to set up an appointment or just drop by (Science Center 341).

Course assistant: Jeffrey Kuan (jkuan@math.harvard.edu).

Lecture notes available here, weekly updated. Please let me know about the inaccuracies and typos you will certainly find.

Course description: an introduction to random matrices, emphasizing how related statistics occur in analytic number theory (spacings between large zeros, moments, low-lying zeros of L-functions).

Prerequisites: probability theory and complex analysis.

Textbooks: for the random matrices part, a reference is Anderson, Guionnet, Zeitouni, An introduction to random matrices. For the number theoretic part, a reference is Montgomery, Vaughan, Multiplicative Number Theory I: Classical Theory.

A tentative schedule for this course is:

• Sep. 4. An introduction on the analogies between random spectra and statistics of zeros of L-functions.
• Sep. 6. Eigenvalues density for the Gaussian ensembles.
• Sep. 11. Macroscopic asymptotics: the semicircle law.
• Sep. 13. Determinantal point processes, Gaudin's lemma.
• Sep. 18. Microscopic repulsion of random eigenvalues.
• Sep. 20. Weyl's integration formula for compact groups and microscopic repulsion of random eigenvalues.
• Sep. 25. The Montgomery-Vaughan inequality and the large sieve.
• Sep. 27. The Riemann ζ function: the functional equation, Weil's explicit formula.
• Oct. 2. Microscopic repulsion of the ζ zeros, by Montgomery.
• Oct. 4. Higher order correlations for the ζ zeros, by Rudnick and Sarnak.
• Oct. 9. The Hardy-Littlewood conjectures and consequences.
• Oct. 11. Linear statistics of random eigenvalues from the Gaussian ensembles
• Oct. 16. Linear statistics of random eigenvalues from the compact groups.
• Oct. 18. Selberg's mollification
• Oct. 23. Selberg's central limit theorem
• Oct. 25. Linear statistics of the ζ zeros.
• Oct. 30. Small ζ moments along the critical axis.
• Nov. 1. The characteristic polynomial of random matrices.
• Nov. 6. The ζ moments, by Keating and Snaith.
• Nov. 8. Models for the ζ moments.
• Nov. 13. Central values of L-functions.
• Nov. 15. Low-lying zeros of families of L-functions
• Nov. 20. A family of symplectic type: Dirichlet L-functions associated to real, quadratic characters.
• Nov. 27. A family of orthogonal type: elliptic curve L-functions.
• Nov. 29. Statistics of zeros for L-functions of curves over finite fields, by Katz and Sarnak, I
• Dec. 4. Statistics of zeros for L-functions of curves over finite fields, by Katz and Sarnak II