MIT PDE/Analysis Seminar, Fall 2001

At MIT in Building 2
In Room 2-147 at 4:30 PM on Wednesdays, unless otherwise noted.

• September 12: David Jerison (MIT)
  ``Regularity of energy minimizing free boundaries in 3 dimensions''

  
  

• September 19: Richard Melrose (MIT)
  ``Scattering by potentials which are Morse at infinity''

  
  

• September 26: Andras Vasy (MIT)
  ``The spectral shift function in many-body scattering''

  
  

• October 3: Mikhail Shubin (Northeastern)
  ``Discreteness of spectrum and strict positivity criteria for magnetic Schrödinger operators''

  
  

• October 10: Maciej Zworski (UC Berkeley)
  ``Pseudospectra of (pseudo)differential operators''

  
  

• October 17: Robert Brooks (Technion/MIT)
  ``The Cheeger constant of a Riemann surface''

  
  Abstract: According to a famous conjecture of Selberg, if S is a congruence surface, the first eigenvalue of the Laplacian of S is at least 1/4, the largest possible value. We study the analogous question for the Cheeger constant of a Riemann surface, which is closely related to the first eigenvalue. Using ideas from probability theory, we show that the analogue of the Selberg conjecture is false, and that congruence surfaces have Cheeger constants uniformly bounded away from the optimal value.

• October 24: Rafe Mazzeo (Stanford)
  ``Conformal theory of the CMC moduli space''

  
  

• October 31: Vadim Kaloshin (MIT)
  ``Regularity of Whitney's embedding of infinite-dimensional sets into finite dimensional spaces''

  
  Abstract:Consider a compact set X in R^n of Hausdorff dimension D and a generic projection P of R^n to R^k for some n>k>D. A classical Marstrand-Matilla theorem says that HD(P(X))=HD(X) or Hausdorff dimension stays the same.

  Now consider a compact set X in a Hilbert space H. It turns out that preservation of Hausdorff dimension under projections is no longer true. Namely, we construct a compact set X of any HD(X)=D in the real Hilbert space l^2 such that for all linear projections P of X into R^n, any n is, HD(P(X))<2. Extensions of Marstrand-Mattila theorem will be discussed.
  
  

• MONDAY, November 5, 4:30-5:30, Room 2-147: Satoshi Ishikawa (Tokyo University)
  ``Totally geodesic Radon transform on symmetric spaces''

  
  

• November 14: TWO TALKS.

  • 3-4pm: Yuri Berest (Cornell)
    ``On Huygens' principle and lacunas for hyperbolic operators with variable coefficients''

    
    

  • 4:30-5:30pm: Tatiana Toro (U Washington/Harvard)
    ``Free boundary regularity for the Poisson kernel below the continuous threshold''

    
    

• MONDAY, November 26: Yakar Kannai (Weizmann Institute)
  ``Sub-Riemannian geometry and kernels associated with certain hypoelliptic operators''

  
  Abstract: The classical calculus of pseudo-differential operators is not suitable for many degenerate operators (such as the Laplacian on the Heisenberg group) which are hypo-elliptic without being elliptic. Using appropriate complex valued solutions ("actions") of a corresponding Hamiltonian system, one may write explicit kernels for resolvents, heat operators, and wave operators. The sub-Riemannian geometry enters in ways quite unlike the Riemannian metrics appearing in the elliptic cases (joint work with R.Beals, B.Gaveau, P. Greiner, and D. Holcman).

• December 5: Paul Kirk (Indiana University)
  ``Topological aspects of Spectral flow''

  
  

A taste of next semester:

• February 6: Gigliola Staffilani (Brown/Harvard/Stanford)

• February 13: Dmitry Dolgopyat (Penn State University)

• March 13: Horng-Tzer Yau (NYU)

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