Analysis-Geometry Seminar 1998-9

Archived Talks from 1998-9:

• Friday, April 30, 1999
  Speaker: Gennady Kasparov (Marseille, visiting Dartmouth College)
  Title: Bolic spaces, operator algebras, and the Novikov conjecture
  Abstract: Bolic metric spaces are generalizations of hyperbolic metric spaces of M. Gromov. They were defined (in my joint work with G. Skandalis) for the purpose of proving the Novikov conjecture on higher signatures of closed smooth manifolds. The result of this work is that the Novikov conjecture is true if the fundamental group of the manifold admits a proper isometric action on a bolic metric space of bounded coarse geometry.
     Large classes of metric spaces, such as non-positively curved simply connected Riemannian manifolds, Euclidean buildings, hyperbolic spaces, are bolic. This allows to give a proof of the Novikov conjecture valid in a number of cases which previously required to be treated separately.

• Friday, April 23, 1999 (No lunch this day; perhaps coffee afterwrds)
  Speaker: Gabriel Katz (Education Development Center & Harvard Graduate School of Education)
  Title: Geometry of Harmonic Forms and Near-minimal Singular Foliations
  Abstract: I shall discuss the restrictions imposed on the singularities of a closed differential 1-form by its harmonicity; the restrictions that are not imposed by the topology of the manifold on which the form lives.
     Then I will comment on the following observation. A simple geometric property (call it A) implies two quite different phenomena: the first is an intrinsic harmonicity of a closed 1-form w and the second is an intrinsic minimality of the foliation Fw defined by the form. The first implication of A was discovered by Calabi, the second, for non-singular foliations, by Sullivan. However, closed forms typically do have singularities and so do the corresponding foliations. Therefore, it is natural to ask whether in the presence of property A, for a closed 1-form w with the Morse type singularities, there exists a metric in which w is harmonic and the leaves of the foliation Fw are the volume-minimizing hypersurfaces. I will show that one can find a metric which harmonizes w and almost minimizes Fw, thus bringing the Calabi's and Sullivan's theories under a single roof. This will imply a number of results about the volume-minimizing cycles in a given homology class.

• 1:30-2:30 Monday, March 29, 1999 (Note unusual day & time!)
  Speaker: Alexander Brenner (Technion, Haifa)
  Title: Polyparabolic Equations, Problems with Parameters, Multiple Operator Zeta Functions, and Related Spectral Asymptotics
  Abstract: We generalize the notion of the parabolicity to the case of multidimensional time equations (i.e. introduce the polyparabolic problems) and obtain the uniqueness and existence theory in the corresponding Sobolev-like scale of spaces.
  On our way to polyparabolic problems we study elliptic boundary value problems with parameters in a bounded domain. The notion of the ellipticity with parameters for differential and pseudodifferential operator pencils can be applied to the resolvent construction which leads to the definition of the corresponding complex powers and zeta - function with polar sets of the first order, the corresponding holomorphic residue-forms and, finally, the two-parameter spectral asymptotics.

• March 12, 1999
  Speaker: Dmitri Vassiliev (Univ of Sussex)
  Title: The Dirac equation without spinors
  Abstract: The talk is an attempt at analysing the geometric meaning of the Dirac equation. It is shown that one can achieve a sensible geometric interpretation by adding an extra time-like coordinate in the target space ("fifth dimension"). This additional degree of freedom may be viewed as the normal displacement of space-time. In other words, in this model space-time is allowed to bend.

• September 25, 1998
  Speaker: Alexander Kozhevnikov (University of Haifa, Israel)
  Title: Parameter-ellipticity for mixed-order elliptic system and applications
  Abstract: Parameter-ellipticity condition for fixed-order elliptic boundary-value problems is known also as the Agmon-Agranovich-Vishik condition. Under this condition the resolvent can be constructed for large enough values of spectral parameter. A generalization of the condition to mixed-order (Douglis-Nirenberg) boundary problems as well as a time-dependent system describing compressible fluid is the main topic of the talk.