# MATH 741: Spectral geometry of random metrics

**INSTRUCTOR:**

D. Jakobson

Office: BH1212

Office Hours: TBA

Tel: 398-3828

E-mail: jakobson@math.mcgill.ca

Web Page: www.math.mcgill.ca/jakobson

Course page: http://www.math.mcgill.ca/jakobson/courses/math741.html

**LECTURES:**

**COURSE DESCRIPTION:**

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. C^{k}, 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.

**Presentations, Grading**

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

## Links

** Curvature of random metrics**
** Introduction to Riemannian Geometry**

**Laplacian, heat kernel etc **

**Comparison Geometry**

**Probability and Geometry**

**Curvature**

**Scalar Curvature**

**Extremal Metrics**

**Generic Metrics**

**Quantum gravity and KPZ conjecture**

**Random waves**

**Spaces of Riemannian metrics and structures on them**

**Spaces of mappings**

**Introductory Links in Differential Geometry, Spectral Geometry etc **

**NOTICE:**
McGill University values academic integrity. Therefore, all
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).

**NOTICE:**
In the event of extraordinary circumstances beyond the University's
control, the content and/or evaluation scheme in this course is
subject to change.