# MATH 599: Topics in Geometry and Topology

** COURSE PAGE:**
Course page: http://www.math.mcgill.ca/jakobson/courses/math599.html

**INSTRUCTORs:**

**LECTURES:**

The course is given in conjunction with
Thematic semester
on Probabilistic Methods in Geometry, Topology, PDE and Spectral
Theory at CRM.

**COURSE DESCRIPTION:**

In the beginning, we shall give a rapid introduction to spectral theory
of the Laplacian on Riemannian manifolds and to infinite dimensional Gaussian
measures.

Next, we shall discuss several examples of applications of probabilistic
techniques in Geometry, Analysis and PDE.

** LECTURE NOTES AND SLIDES**

** POSSIBLE TOPICS:**

We shall attempt to make the course self-contained. Advanced undergraduate
students and graduate students are welcome!

**Presentations, Grading**

The students will be expected to choose a topic for a short
presentation in class, in consultation with instructors, and to give
a 30-40 minute talk on that topic.

The students may also write a short report related to the subject of their
talk.

The grades will be based on the presentation (and on the report, if the
student chooses to write it).
** Possible themes for Presentation**

Any topic in Analysis, Geometry, PDE or Probability related to the
course, to be discussed with instructors

## Previous courses on related topics at McGill

Math 741, Spectral geometry
of random metrics

## Dmitry Jakobson's lecture at Fields institute

A lecture on Quantum Chaos etc at the Fields Medal Symposium:
interactive, and
static

## Selected slides from talks at CRM, Fall 2016

Random eigenfunctions:
I. Wigman;
A. Taylor;
V. Cammarota;
M. Rossi.
Quantum ergodicity for random bases/operators:
R. Chang;
E. Le Masson.
## Probability and Geometry

A book by R. Adler and J. Taylor titled "Random fields and Geometry"
can be found on Robert Adler's
publications page
Jonathan Taylor's web
page
Stanislav Molchanov's
web page

## Background material (from Math 741 links and other sources)

** Spectral theory of the Laplacian on Riemannian manifolds**
Yaiza Canzani's
Lecture
notes from her 2013 course at Harvard.
** Quantum Chaos**
P. Sarnak:
Arithmetic Quantum Chaos,
"Mass equidistribution and zeros/nodal domains of modular forms" slides of
Dartmuth lectures, July 2010
A. Gorodnik:
Dynamics and Quantum Chaos on hyperbolic surfaces, a course at Bristol
** Random Matrices**
T. Pereira
V. Kargin and E. Yudovina
G.W. Anderson, A. Guionnet and O. Zeitouni
F. Rezakhanlou
** Sobolev spaces, Sobolev embedding theorems**
A. Benyi and T. Oh,
The Sobolev inequality on the torus revisited,
Publ. Math. Debrecen.
Terry Tao, UCLA,
Notes on Sobolev spaces
L. Chen, UCDavis, 2011:
summary
J. Viaclovsky, MIT, 2004:
Lecture notes. See lectures 16, 17, 18.
Manifold version:
J. Kelliher, UC Riverside (d'apres Aubin).
** Introduction to Riemannian Geometry**
Sigmundur Gudmundsson's
lectures notes, especially chapters 6, 7, 8, 9.
**Laplacian, heat kernel etc **

Lectures on semiclassical analysis by M. Zworski.
A
future book by Victor Ivrii (large file!)
There are many lectures notes on the
home page
of Robert Brooks
Notes on heat kernel asymptotics by D. Grieser
Lecture Notes by Melrose
P. Gilkey: Invariance theory, the heat equation, and the
Atiyah-Singer index theorem:
EMIS server
and
pdf file
P. Gilkey, J. Leahy and J. Park: Spinors, spectral geometry, and
Riemannian submersions:
EMIS server
Lecture Notes by Melrose
**Comparison Geometry**

MSRI Publications, Volume 30: conference proceedings, edited by
Karsten Grove and Peter Petersen.
Cheeger and Ebin, "Comparison Theorems in Riemannian Geometry"
link
A web
page about comparison geometry by Terry Tao
Wolgang Meyer,
Lecture
notes on Toponogov's theorem;
conference in honor of Toponogov's 70th birthday, 2000.
**Probability and Geometry**

A book by R. Adler and J. Taylor titled "Random fields and Geometry"
can be found on Robert Adler's
publications page
Jonathan Taylor's web
page
Stanislav Molchanov's web page
**Curvature**

Gromov's lecture "Sign and geometric meaning of curvature:"
Milan
journal and
another link
Jeff Viaclovsky's
Lecture Notes in Riemannian Geometry
**Scalar Curvature**

Kazdan and Warner,
Scalar curvature and conformal deformation of Riemannian structure
J. Rosenberg,
Manifolds of positive scalar curvature: a progress report
**Generic Metrics**

Karen Uhlenbeck's paper
Generic properties
of eigenfunctions, American Journal of Mathematics, 1976.
Lecture notes on generic
metrics that I gave 2 years ago, written up and typed by Michael
McBreen. **Note:** This is an overview, few detailed proofs
are given.
**Quantum gravity and KPZ conjecture**

B. Duplantier and S. Sheffield:
Liouville Quantum Gravity and KPZ
B. Duplantier and S. Sheffield:
Duality and KPZ in Liouville Quantum Gravity
X. Xu, J. Miller and Y. Peres:
Thick Points of the Gaussian Free Field
S. Sheffield:
Gaussian free fields for mathematicians
E. D'Hoker and D. Phong:
The geometry of string perturbation theory
**Random waves**

J.-P. Kahane:
Some random series of functions, 2nd edition
S. Zelditch:
Real and complex zeros of Riemannian random waves
F. Nazarov and M. Sodin:
On the Number of Nodal
Domains of Random Spherical Harmonics
J. Toth and I. Wigman
Counting open nodal lines of random waves on planar domains
D. Hejhal and B. Rackner:
On the topography of Maass waveforms for PSL(2,Z)
I. Wigman:
Fluctuations of the nodal length of random spherical harmonics
R. Aurich, A. Backer, R. Schubert and M. Taglieber.
Maximum norms of chaotic quantum eigenfunctions and random
waves
**Spaces of Riemannian metrics and structures on them**

M. Berger and D. Ebin: Some
decompositions of the space of symmetric tensors on a Riemannian manifold
N. Smolentsev:
Spaces of Riemannian metrics
O. Gil-Medrano and P. Michor:
The
Riemannian manifold of all Riemannian metrics
D. Freed and D. Groisser:
The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group
A. Fischer and J. Marsden:
The manifold of conformally
equivalent metrics
Brian Clarke: Thesis
and papers: The
Completion of the Manifold of Riemannian Metrics,
The Metric Geometry
of the Manifold of Riemannian Metrics over a Closed Manifold, and
The Riemannian L2
topology on the manifold of Riemannian metrics.
**Spaces of mappings**
F. Morgan:
Measures
on spaces of surfaces
P. Michor and D. Mumford:
Riemannian geometries on spaces of plane curves and
An overview of the Riemannian metrics on
spaces of curves using the Hamiltonian approach

**Introductory Links in Differential Geometry, Spectral Geometry etc **
some cool graphics: GANG;
Plane curves
Lecture notes in DG on the net:
Nigel Hitchin;
Gabriel Lugo;
Balazs Csikos;
C.T.J. Dodson
Differential Geometry pages at
wikipedia
and at
mathworld
Survey papers from Alan Weinstein's course at Berkeley:
page 1 and
page 2
Survey papers from a course by Tamas Hausel at UTexas; in particular
Eigenvalues and the Heat Kernel by A. Young;
Curvature and
fundamental group by S. Kang.
UChicago
warmup page

**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 accord with McGill University's Charter of Student
Rights, students in this course have the right to submit in English or
in French any work that is to be graded.

**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