Analysis-Geometry Seminar 1997-8

Archived Talks from 1997-8:

• December 5, 1997
  Speaker: Maxim Braverman (Hebrew University, Jerusalem)
  Title: Equivariant Mayer-Vietoris sequence in Dolbeault cohomology
  Abstract: Assume that the circle acts holomorphically on a compact Kaehler manifold M. The construction of E.Lerman allows to cut M into two pieces which intersect by the symplectic reduction. We consider the union M' of those two pieces as a singular complex manifold and show that this manifold may be viewed as a deformation of the original manifold M. In particular, the twisted Dolbeault cohomology of M may be estimated by the twisted Dolbeault cohomology of M'. Moreover, the Dolbeault cohomology of M with coefficients in a pre-quantum line bundle is equivariantly isomorphic to the corresponding cohomology of M'. That, in particular, proves the conjecture of Wu and Zhang about existence of an equivariant Mayer-Vietoris-type exact sequence for symplectic cuttings.

• November 21, 1997
  Speaker: Jim Propp (MIT)
  Title: Boundary Effects for Dimer Models
  Abstract: Dimer models of statistical mechanical systems on finite bipartite graphs in two dimensions can exhibit surprising effects, associated with the existence of the height representation for dimer configurations. In such cases, microstatistics and local entropy can vary over macroscopic distances, giving rise to spontaneously-formed domains; boundaries between domains appear to correspond to phase-transitions of the model in the presence of a suitably defined external field. (Note: All of the phrases used in this abstract will be defined in the lecture.)

• November 14, 1997
  Speaker: Henri Moscovici (Ohio-State University)
  Title: Vafa-Witten Inequalities and Poincare Duality in K-theory
  Abstract. We shall explain how to derive the Vafa-Witten uniform upper bounds for the first eigenvalues from first topological principles, in the general framework of noncommutative geometry. We shall then give examples of noncommutative spaces that satisfy the Vafa-Witten inequalities for the first eigenvalues but not the inequalities for higher eigenvalues.

• November 7, 1997
  Speaker: Mikhail Shubin (Northeastern University)
  Title: Elliptic Boundary Value Problems on Covering Manifolds
  Abstract. We consider elliptic boundary value problems on non-compact covering manifolds with coefficients which are periodic i.e. invariant under the deck transformations. The corresponding operators in uniform Sobolev spaces are Fredholm in the sense of the Breuer version of the Fredholm theory in von Neumann algebras (but not in the usual sense). We will discuss analytic consequences of this property and open questions.

• October 31, 1997
  Speaker: P.Robert Kotiuga (Boston University, Dept. Elec. & Comp.Eng.)
  Title: Whitney Form Finite Element Discretizations of Abelian Chern-Simmons Functionals and Applications
  Abstract: This talk considers Chern-Simmons functionals in the context of computational electromagnetics. Nonlinear magnetostatics and impedance imaging in three dimensions provide the backdrop, and in center stage is finite element discretization by means of Whitney forms (i.e. (p-l) differential forms with key functorial properties.) These discretizations are exceptional for two reasons:
  1. The condition number is approximately the square root of that of the discretization of the Laplace-Beltrami operator since it is an elliptic operator of the Dirac type.
  2. The part of the discrete quadratic form corresponding to the Chern-Simmons term involves no metric information, even on nonuniform meshes! This follows from the naturality of Whitney forms.
     
  Next, the determinant of the quadratic form is considered in relation to the original functional determinant. This provides a new perspective on both the equality of Reidemeister and analytic torsions, and topological quantum field theories where Wilson loop observables are related to linking numbers.
  We conclude with open problems concerning manifolds with boundary which have implications for both torsion invariants and impedance imaging.

• October 24, 1997
  Speaker: Dimitry Leites (Stockholm University)
  Title: Supersymmetry of the Schrodinger and Korteweg - de Vries operators
  Abstract: In the 1970's, A.A.Kirillov discovered a supersymmetry of the Schrodinger and Korteweg - de Vries operators. We superize Kirillov's results still further to embrace the Khesin-Malikov generalization of the Drinfeld-Sokolov reduction to the pseudodifferential operators.

• October 10, 1997
  Speaker: David Adams (Trinity College, Dublin)
  Title: Discrete analogues of formulae in Hodge-de Rham theory and discretisation of topological field theory
  

• October 3, 1997
  Speaker: Vadim Tkachenko (Ben Gurion University of the Negev)
  Title: Periodic Dirac operators with skew-symmetric potential matrices
  Abstract: We describe the monodromy matrices of the operators from the title and give applications to the characterization of the typical spectra. We prove the density of the potential matrices of finite band operators in the space of all skew-symmetric matrices.

• Monday, September 22, 1997
  (Joint with Geometry-Algebra-Singularities Seminar)
  Speaker: Michael Farber (Tel Aviv University) (Tel Aviv University, Israel)
  Title: Euler structures, Spin^c-structures, and refined Reidemeister type torsion invariants of manifolds
  Abstract: In the talk I will describe:
  1. The torsion invariant of Turaev depending on the choice of an Euler structure, or a Spin^c-structure;
  2. The other combinatorial torsion invariant computing the Ray-Singer analytic torsion;
  3. The relation between 1. and 2. involving characteristic classes.

• September 19, 1997
  Speaker: Jochen Bruening (Berlin)
  Title: Heat asymptotics for general boundary value problems
  Abstract: We consider boundary value problems for Dirac type operators and their Laplacians which generalize the well known spectral boundary condition of Atiyah, Patodi, and Singer. We determine a large class of such conditions which admit heat asymptotics and point out some applications.