The first series of Chern Lecturers in 2022-23 will be given by Professor Peter Sarnak, Professor, Princeton University and Institute for Advanced Study (January 31st, February 2nd, February 7th, and February 8th, 2023). The title of the series is "Spectra of locally uniform geometries".
Lecture 1 of 4 (Jan-31-2023 @ 4:10–5:00 pm, UC Berkeley Campus, Location: Bechtel Sibley Auditorium): Spectra of Locally Uniform Geometries: Prescribing the spectrum --- Locally symmetric Riemannian spaces. ABSTRACT: We review recent developments (conformal bootstrap and random covers) concerning the bass-note of the spectrum of Laplacians on hyperbolic manifolds and on large cubic graphs. We highlight rigidity to creating spectral gaps when restricting the geometries. Lecture 1 will focus on locally symmetric Riemannian spaces.
Lecture 2 of 4: (Feb-02-2023 @ 4:10–5:00 pm, Location: 60 Evans Hall): Spectra of Locally Uniform Geometries: Prescribing the spectrum --- Cubic graphs. ABSTRACT: We review recent developments (conformal bootstrap and random covers) concerning the bass-note of the spectrum of Laplacians on hyperbolic manifolds and on large cubic graphs. We highlight rigidity to creating spectral gaps when restricting the geometries. Lecture 2 will focus on cubic-graphs and the eigenvalues of Frobenius for curves and abelian varieties. Joint work with Alicia Kollar and Fan Wei. Note: This lecture will be taking place as part of the Department of Mathematics' Spring '23 Colloquium Series.
Lecture 3 of 4 (Feb-07-2023 @ 4:10–5:00 pm, UC Berkeley Campus, Location: Bechtel Sibley Auditorium): Spectra of Locally Uniform Geometries: The additive structure of the spectrum of a metric graph. ABSTRACT: Metric graphs are compact one dimensional Riemannian manifolds with singularities and have been studied in different guises. Using tools from diophantine analysis we determine the additive and transcendence properties of their spectra and its applications to crystalline measures. Joint work with Pavel Kurasov.
Lecture 4 of 4: (Feb-08-2023 @ 4:10-5:00 pm, UC Berkeley Campus, Location: 1015 Evans): Spectra of Locally Uniform Geometries: The general Ramanujan. and density conjectures. ABSTRACT: The naive extension of Satake's formulation of the Ramanujan conjectures fails in any generality. Arthur's conjectures are a far reaching remedy but they remain largely out of reach. After reviewing these briefly we describe a simple density conjecture which serves as complete substitute for various applications and which has been proven recently in many cases.