Organized by: G. Grubb and Jan Philip Solovej
The workshop is taking place January 21 - 23, 1999, at the Mathematics Department, Copenhagen University.
All interested colleagues and students are welcome. Please inform us if you plan to attend the workshop and (before Jan. 14) whether you want to participate in lunch on Friday and/or Saturday (estimated price 50 DKK). We may also be able to help with hotel reservations if asked well in advance.
Information about the hotels and directions to the Ørsted Institut
The workshop is supported by the Danish Network on PDE, Analysis and Applications (under the The Danish Natural Science Research Council) and by MaPhySto, Centre for Mathematical Physics and Stocastics, (under the Danish National Research Foundation).
Title: Signature formulae for singular spaces
Abstract: There have been many attempts to extend the famous Hirzebruch Signature Theorem, or various aspects of it, to compact stratified spaces like pseudo manifolds or algebraic varieties. What has been missing so far is an analog of Hirzebruch's local index formula. In this talk (which presents joint work with Robert Seeley) we give such a formula for the case of Witt spaces with one singular stratum and a suitable metric.
Title: Harmonic maps between singular Riemannian spaces
Peter B. Gilkey
Title: Heat Asymptotics with spectral boundary conditions
Abstract: Let be an elliptic complex of Dirac type on a compact Riemannian manifold with smooth boundary. We impose spectral boundary conditions and determine formulas for the asymptotic coefficients a0, a1, and a2 for the asymptotic expansion of the heat trace of the associated operators of Laplace type. We do not assume that the structures are product near the boundary. This is joint work with S. Dowker and K. Kirsten.
Title: On the semi-classical expansion of the thermodynamic limit associated with the ground-state energy of a Kac operator.
Title: Local P-convexity.
Title: On Witten-Laplacians and a covariance formula from statistical mechanics.
Abstract: The subject is a formula which was introduced in statistical mechanics in recent years by B. Helffer and J. Sjöstrand in connection with phase transitions; it provides a rewriting of the covariance of two functions on a Euclidean space. One purpose of the talk is to explain how one in the proof of this formula is led immediately to a fine version of the Closed Range Theorem in Hilbert space theory and to, for example, an analysis of the Witten-Laplacians by means of the Weyl calculus of pseudo-differential operators. (Although there is a simple heuristic argument for the formula, the proof of it turns out to be either trivial (in certain cases) or to require advanced tools--seemingly without any `soft transition' between these two alternatives.) The talk ends with a discussion of the explicit criteria (to be imposed on the probability measure's density function), which have been shown to yield the formula.
Title: Conditional trace of commutators of Pseudodifferential operators.
Abstract: We define a conditional trace of the commutator of two pseudodifferential operators (psdo). If the commutator is not of the trace class, then its conditional trace is not equal to zero, but leads, however, to some expressions devined invariantly. We show how the conditonal trace is related to the Guillemin-Wodzicki noncommutative residue.
Title: Nonlinear gluing
Abstract: I will survey some recent advances in the theory of minimal and constant mean curvature surfaces, with particular emphasis on a new result, jointly with Frank Pacard, about geometric connected sums of such surfaces. The main emphasis will be the description of a new and very simple technique for gluing theorems of this sort.
Title: Backscattering and nonlinear Radon transformation
Title: Fibred boundaries, complete manifolds and propagation of singularities
Title: Local formula for indices of Fourier Integral Operators
Abstract: We give local formulae for index of an elliptic Fourier operator associated to a contact diffeomorphism between cosphere bundles of two compact Riemannian manifolds, in particular generalising the results of Epstein and Melrose. The proof is based on an algebraic index theorem for symplectic Lie algebroids.
Title:Noncommutative Residues and Heat Trace Asymptotics for Boundary Value Problems
Abstract: The noncommutative residue was discovered by M. Wodzicki in 1984 and, in a slightly different context, by V. Guillemin in 1985. Formally, it is a trace on the algebra of pseudodifferential operators on a compact manifold; in fact, it is the unique trace on the symbol algebra up to multiples. The noncommutative residue has meanwhile found a wide range of applications both in mathematics and in mathematical physics, ranging from spectral theory to noncommutative geometry and gravitation. Here, the focus is on manifolds with boundary. It could be shown that there is an essentially unique trace on the algebra of all boundary value problems in Boutet de Monvel's calculus which extends the trace of Wodzicki (Fedosov, Golse, Leichtnam, Schrohe 1994). While this description was purely in terms of the associated symbols, recent joint work with G. Grubb shows that the noncommutative residue can also be recovered from suitable heat trace expansions.
Title: Resonances and bottles
Abstract: We discuss estimates for the number of resonances near the real axis in bottle-like situations. The proof uses a variant of a recent local trace formula and an elaboration of an argument of Carleman-Vodev.
Title: Asymptotic expansions of heat kernels and conformal geometry.
Abstract: In this talk we shall study natural elliptic operators on compact Riemannian manifolds and survey some of the local and global invariants associated with the corresponding heat equation. In particular we will give the relation between regularized zeta determinants and conformal variation of the metric, leading to explicit formulas in low dimension. Finally we give some of the extremal properties of these determinants as well as a discussion of the equivariant case.