MATH 741: Spectral geometry of random metrics
Office Hours: TBA
Web Page: www.math.mcgill.ca/jakobson
Course page: http://www.math.mcgill.ca/jakobson/courses/math741.html
We shall discuss geometric and spectral properties of random Riemannian
metrics on a compact manifold lying in a fixed conformal class. We shall
start our discussion with measures on spaces of metrics such that the
typical metrics are a.s. Ck, and aim to eventually understand
the measures for which the typical metrics are less regular (e.g. like
those in 2-dimensional quantum gravity). We shall study various geometric
and spectral functionals, related to curvature, isoperimetric constants,
spectra and eigenfunctions of Laplacians and other elliptic operators.
We may also explore connections to conformal field theory, quantum
gravity, random wave model in quantum chaos, and the theory of Gaussian
random fields on manifolds, as time permits.
The students registered for the course will be expected to make
one or two oral presentation (30-45 minutes) on one of the topics
suggested by the instructor. The grades will be based
on these presentations.
Possible themes for Presentation
Lecture notes from previous courses at McGill
Curvature of random metrics
Introduction to Riemannian Geometry
Laplacian, heat kernel etc
Probability and Geometry
Quantum gravity and KPZ conjecture
Spaces of Riemannian metrics and structures on them
Spaces of mappings
Introductory Links in Differential Geometry, Spectral Geometry etc
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students must understand the meaning and consequences of
cheating, plagiarism and other academic offences under the Code
of Student Conduct and Disciplinary Procedures (see McGill web page
on Academic Integrity
for more information).
In the event of extraordinary circumstances beyond the University's
control, the content and/or evaluation scheme in this course is
subject to change.