MAT1846HS · Length and Laplace Rigidity

Mon 11:00AM - 12:00PM
Wed 11:00AM - 1:00PM
Office Hours
Wed 1:00PM – 2:00PM

Important information · zoom link

Due to the December 2021 Covid Omicron surge, classes will be online until (at least) January 31 2022. Lectures will be delivered by Zoom; please join this zoom link.
Should you have any questions, please send me an email message.


There is a Quercus page set up for this course; if you plan to attend this course without enrolling please send me an email and I can add you to Quercus. It will make my life easier to distribute announcements and other important information. I will keep the material available in this page for reference.


The syllabus can be downloaded here Light Dark


January 10
  General information, Course Introduction
January 12
  Trace Formulas Preliminaries
January 12
  Trace Formulas Preliminaries
January 17
  The Poisson's summation formula
January 19
  Heat trace on tori
January 24
  Wave trace on tori
January 26
  Weyl's law I
January 31
  Weyl's law II
February 2
  Trace formulas and oscillatory integrals
February 7
  Wave trace, wrapping up
February 9
  Anosov flows and closing lemma
February 14
  Closing Lemma, Livšic's Theorem
February 16
  Cone Hyperbolic flows, Jacobi fields
February 28
  Hyperbolicity of Geodesic flow in negative curvature
March 2
  Transitivity of Geodesic flows in negative curvature I
March 7
  Transitivity of Geodesic flows in negative curvature II
March 9
  Transitivity of Geodesic flows in negative curvature III
March 14
  Proof of Guillemin–Kazhdan I
March 16
  Proof of Guillemin–Kazhdan II
March 23
  Proof of Otal's Theorem I
March 28
  Proof of Otal's Theorem II
March 30
  Proof of Otal's Theorem III
April 4
  Sunada Construction I
April 6
  Sunada Construction II

List of reading items

A. Uribe – Trace Formulae
J. Chazarain - Formule de Poisson pour les variétés riemanniennes
Amie Wilkinson – Lectures on Otal's Theorem
J.-P. Otal – The length spectrum of negatively curved manifolds (in French)
V. Guillemin, D. Kazhdan – Some inverse spectral results for negatively curved 2-manifolds
T. Sunada (砂田 利一) – Riemannian coverings and isospectral manifolds
R. Brooks - The Sunada Method
S. Zelditch - Inverse spectral problem for analytic domains II. ℤ₂–symmetric domains
S. Zelditch - Spectral determination of analytic bi-axisymmetric plane domains
C. Guillarmou, T. Lefeuvre - The marked length spectrum of Anosov manifolds
—, V. Kaloshin, Qiaoling Wei - Dynamical spectral rigidity among ℤ₂–symmetric strictly convex domains close to a circle