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:
  • Organizational meeting: Wednesday, January 13, 12:30, Burnside 1120.
  • Monday, 12:30-14:00, Burnside 1120.
  • The second lecture will alternate between wednesday and friday.
  • Wednesdays, 12:30-13:30, Burnside 1120 (no lecture on jan. 20)
  • Fridays, 12:00-13:30, Burnside 1214
  • Week of January 18: monday (jan. 18) and friday (jan. 22)
  • Lecture on january 22 was cancelled due to family emergency.
  • Week of January 25: monday (jan. 25) and wednesday (jan. 27)
  • Week of February 1: monday (feb. 1) and wednesday (feb. 3)
  • Week of February 8: monday (feb. 8) and wednesday (feb. 10)
  • Week of February 15: monday (feb. 15) and wednesday (feb. 17)
  • Week of March 15: monday (march 15), 12:30-14:00, room 1120; and friday (march 19), 12:00-13:30, room 1214.
    • 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. 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.
    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

  • Any topic in analysis, geometry or probability related to the course, to be discussed with instructor

    • Lecture notes from previous courses at McGill

  • Sebastein Bacle's Masters Thesis that contains lecture notes from a previous course, Extremal metrics in Geometry and Graph Theory that I taught in 2004.

    • Links

    Curvature of random metrics
  • Y. Canzani, D. Jakobson and I. Wigman. "Scalar curvature and Q-curvature of random metrics." Short announcement: pdf; long version: pdf.

    • 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
  • to be continued

    • 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

    • Extremal Metrics

  • El Soufi and Ilias: Paper 1 and Paper 2
  • Paper by Ros
  • Paper by Jakobson, Nadirashvili and Polterovich
  • Paper by Calabi
  • Lecture notes by Alice Chang (see also her homepage)

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