MATH 667: Seminar in Spectral Theory II

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  • D. Jakobson
    Office: McGill, Burnside Hall, 1220
    Office Hours: Wed, 10:30-11:30 and 12:30-13:30
    Tel: 398-3828
  • Iosif Polterovich
    Office: Universite de Montreal, Pav. Andre-Aisenstadt, 5299
    Tel: 343-5899

  • Thursday, 13:30-14:30, (alternating between McGill and Universite de Montreal)
  • Room, Universite de Montreal: Pav. Andre Aisenstadt, 5448.
  • Room, McGill: Burnside 1120.
  • First lecture: January 13, Universite de Montreal.
    Lecturer: Iosif Polterovich
    Title: How large can the smallest eigenvalue of the Laplacian be?
  • January 20, McGill.
    Speakers: G. Roy-Fortin and G. Poliquin.
    Title: Optimization of higher laplacian eigenvalues for the planar Dirichlet and Neumann problems.
  • January 27, Universite de Montreal.
    Speaker: D. Jakobson
    Title: Limits of eigenfunctions on arithmetic flat tori.
    Note that there will be a related talk by Fabricio Macia in the analysis seminar on Friday, January 28, at McGill in Burnside 920, from 14:30-15:30.
  • February 3, UdeM.
    Speaker: T. Aissiou
    Title: L^p norms of eigenfunctions on arithmetic flat tori.
  • February 10, McGill.
    Speaker: Y. Canzani
    Title: Scalar curvature for random metrics.
  • February 17, Universite de Montreal.
    Speaker: Guillaume Lavoie
    Title: Asymptotic behavior of eigenfunctions on the disk.
  • February 24: Spring break.
  • March 3, Universite de Montreal.
    Speaker: Janine Bachrachas
    On mean curvature flow.
  • March 10: University of Montreal
    Speaker: Alexei V. Penskoi (Moscow State University, Independent University of Moscow and Bauman Moscow State Technical University)
    Title: Extremal spectral properties of Lawson tau-surfaces and the Lame equation
    Abstract: Given a closed compact surface, eigenvalues of the Laplace-Beltrami operator are functionals on the space of Riemannian metrics of fixed area on this surface. The question about extremal metrics for these eigenvalues is a difficult problem of a differential geometry.
    In this talk we shall describe significant advances is this domain happened during last ten years and last results about extremal metrics on Lawson tori and Klein bottles representing an interesting interplay between extremal metrics, minimal surfaces and the classical Lame equation.
  • March 17, McGill.
    Speaker: D. Jakobson
    Eigenfunctions with few critical points
  • March 24, University of Montreal.
    Speaker: Leonid Polterovich (University of Chicago and Tel Aviv University)
    Title: Nodal inequalities on surfaces
  • March 31, McGill.
    Speaker: A. Komech (Texas AM University)
    On spectral stability of nonlinear Dirac equation
    Abstract: We are interested in the spectral stability of solitary wave solutions to the nonlinear Dirac equation in 1D (known as the 1D Soler model, also known as the Gross-Neveu model). That is, we linearize the equation at a solitary wave and examine the presence of eigenvalues with positive real part. We consider the nonlinear Dirac equation from the point of view of Grillakis-Shatah-Strauss formalism and show that the violation of the Vakhitov-Kolokolov stability criterion leads to the bifurcation of unstable eigenvalues (as the frequency changes) from zero. We also analyze the bifurcations from the essential spectrum. We combine these results to prove that small amplitude solitary wave solutions to the nonlinear Dirac equation in 1D are spectrally stable.
  • April 7, U. Montreal
    Speaker: J. Barnes
    On L^p norms of eigenfunctions of Laplacians
  • April 14, McGill
    Speaker: M. Smilovic
    Riemannian geometries on spaces of plane curves
  • April 21 no classes
  • April 27, U. Montreal, 11am-12pm.
    Speaker: E. Lavrova
    Zero set of Fourier transform and spectral geometry
  • May 19, McGill
    Burnside 920 (different room!), 13:30-14:30
    Daniel Peralta Salas (Madrid)
    Topological monsters in PDE
    Abstract: Can the infinite jungle gym be the zero set of a harmonic function? Are there two harmonic functions whose joint zero set contains all knot and link types? In this talk we will show the existence of harmonic functions in R^3 exhibiting these and other topological monsters, keeping technicalities to the bare minimum. This talk will be based on joint work with Alberto Enciso.

    Topics discussed in this course will include nodal geometry of eigenfunctions of elliptic operators, eigenvalue inequalities, spectral asymptotics.
    Presentations, Grading
    The students registered for the course will be expected to make one or two oral presentations (40-50 minutes) on one of the topics suggested by the instructors. The grades will be based on these presentations.

    Math 667, Fall 2010
    Seminar in Spectral Theory I

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