MATH 599: Topics in Geometry and Topology

COURSE PAGE: Course page:
  • Linan Chen
    Office BH1230
    Office Hours: TBA
    Tel: 398-3858
  • Dmitry Jakobson
    Office: BH1220
    Office Hours: TBA
    Tel: 398-3828
    Web Page:

  • Organizational meeting: Thursday, September 8, 13:30, Burnside 1214.
  • Tuesday, 14:30-16:00, Burnside 708 (7th floor).
  • Thursday, 13:30-15:30, Burnside 1214.
  • Friday, 9:00-10:30, Burnside 1120 (for those who could not make it either Tuesday or Thursday)
  • On Thursday, October 27 and November 11, Janosch Ortmann will give 2 lectures on random matrices at Concordia, Room LB 759-6, from 13:30-14:30. On Thursday, November 4, the lecture will be at McGill, as usual.
  • There will be no lecture on Friday, October 28.
  • There will be no lecture on Friday, November 4.

  • The course is given in conjunction with Thematic semester on Probabilistic Methods in Geometry, Topology, PDE and Spectral Theory at CRM.
    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.
  • Linan Chen: September 27
  • Linan Chen: September 29
  • Linan Chen: October 4
  • Linan Chen: October 6
  • Linan Chen: November 1
  • Slides, talk at the 2013 Fields medal symposium at Fields Institute, Toronto (pdf)
  • Nalini Anantharaman, Aisenstadt lectures at CRM, August 2016: lecture 1, lecture 2, lecture 3.
  • Dima Jakobson: Curvature of random metrics and another version
  • Dima Jakobson: Measures on spaces of metrics with the fixed volume form (CMS, 2015)
  • Notes for the lectures on Random Matrices by Janosch Ortmann: Random matrix course by B. Valko at U. Wisconsin, 2009. Scroll down for the links to lecture notes.

  • "Random wave" model in quantum chaos, including results about the rate of growth of L^p norms and maxima of random eigenfunctions, the size and topology of their nodal and level sets, critical points etc:
  • Nazarov and Sodin: Lectures on random nodal portraits (Sodin); arxiv:0706.2409; arxiv:1507.02017
  • Gayet and Welshinger: arxiv:1406.0934; arxiv:1503.01582; arxiv:1605.08605
  • Sarnak, Wigman and Canzani: arxiv:1510.08500; arxiv:1606.05766; arxiv:1412.4437
  • Wigman, Marinucci, Cammarota, Rossi et al: arxiv:1510.00339; arxiv:1504.01868; arxiv:1409.1364
  • Nicolaescu: arxiv:1101.5990; arxiv:1209.0639; arxiv:1509.06200 arxiv:1511.04965
  • Solutions of PDE with random initial conditions:
  • Burq and Lebeau: arxiv:1111.7310
  • Komech, Kopylova, Suhov et al: arxiv:1201.6221; arxiv:0508.4748; arxiv:0508042; arxiv:0508039
  • "Generic" properties of eigenfunctions of elliptic operators (Uhlenbeck, Bando-Urakawa, Albert et al)
  • Uhlenbeck: Generic properties of eigenfunctions, American Journal of Mathematics, 1976.
  • S. Bando and H. Urakawa: Generic properties of the eigenvalue of the Laplacian for compact Riemannian manifolds, Tohoku Mathematical Journal, Second Series, Vol. 35 (1983) No. 2, pages 155-172.
  • J. Albert: Generic properties of eigenfunctions of elliptic PDE Transactions of the AMS Vol. 238 (1978), pages 341-354.
  • Using spectral theory methods to parametrize random maps; applications to constructing probability measures on manifolds of metrics (e.g. conformal classes, manifolds with the fixed volume form etc). Connections to Quantum Gravity.
  • F. Morgan: Measures on spaces of surfaces
  • L. Chen and D. Jakobson. "Gaussian free fields and KPZ relation in R^4." pdf
  • B. Clarke, D. Jakobson, N. Kamran, L. Silberman and J. Taylor. "Manifold of metrics with fixed volume form." With an appendix by Y. Canzani, D. Jakobson and L. Silberman. pdf
  • Applications in geometry, spectral theory, quantum chaos:
  • S. Chatterjee and J. Galkowski: Arbitrarily small perturbations of Laplacian are QUE, arxiv:1603.00597
  • Y. Canzani, D. Jakobson and I. Wigman. "Scalar curvature and Q-curvature of random metrics." pdf.

  • 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