Humboldt Universität zu Berlin
Naturwissenschaftliche Fakultät II
Institut für Mathematik
Prof. J. Brüning

Seminar on Geometric Analysis and Spectral Theory
Sommersemester 2004

The seminar usually takes place every Wednesday during the semester.

The talks will be held at the following location:
Mathematics Institute of the Humboldt University in Berlin-Adlershof, Rudower Chaussee 25, Room I.013.

If you are coming for the first time to Adlershof you might wish to consult a map of Berlin.
The map of the Campus Adlershof and of the Institute building might also be useful.

How to get to Adlershof : take the S-Bahn S9 in the direction Schoenefeld up to Adlershof station.
Our building is situated within a short bus ride (three stations) from the S-Bahnhof.
The bus station Magnus Str. is in front of the Institute.

Frederik Witt (Oxford University):  

21 April, 2004	4:30 pm, Raum 1.013	Dr. Ursula Ludwig (MPI Leipzig)	Morse theory on stratified spaces with tangential conditions
28 April, 2004	4:30 pm, Raum 1.013	Dr. Joa Weber (ETH Zürich)	Floer homology of cotangent bundles and noncontractible periodic orbits
5 May, 2004	4:30 pm, Raum 1.013	Prof. V. A. Geyler (Saransk)	Continuity of the Green functions and the heat kernels for Schrödinger operators on Riemannian manifolds
5 May, 2004	4:30 pm, Raum 1.013	Dr. G. Marinescu (HU)	Compactification of some complete Kähler manifolds with negative curvature and a strongly pseudoconvex end.
5 May, 2004	4:30 pm, Raum 1.013	Dr. D. Pliakis (Uni Mannheim)	Small time heat expansion for analytic hypersurfaces with isolated singularities
16 June, 2004	4:30 pm, Raum 1.013	Dr. Gregor Weingart (z.Zt. U Hamburg)	A Closed Formula for the Heat Kernel Coefficients of Twisted Laplacians
30 June, 2004	4:30 pm, Raum 1.013	Dr. Kai Köhler (U Düsseldorf)	Quaternionische und holomorphe Torsion
7 July, 2004	4:30 pm, Raum 1.013	Dr. D. Grieser (Uni Bonn)	Dirichlet-Spektrum und Streutheorie
7 July, 2004	5:30 pm, Raum 1.013	Dr. Anna Wienhard (Uni Bonn)	Group actions on Hermitian symmetric spaces
7 July, 2004	4:30 pm, Raum 1.013	Dr. Frederik Witt (Oxford University)	Supersymmetries and Hitchin's variational principle in dimensions 7 and 8, Part I

Morse theory on stratified spaces with tangential conditions.

Abstract: We develop a Morse theory on abstract stratified spaces under assumptions which are degenerate in the sense of Goresky/MacPherson. The gradient vectorfield is assumed to be tangential, and the Morse function is increasing normally to the strata. Under these assumptions we can construct the stratified Morse-Witten complex, whose homology is isomorphic to the singular homology of the stratified space (with coefficients in $Z_2$). Unlike to the theory of Goresky/MacPherson, here critical points of a Morse function are related to singular homology (and not to intersection homology).

Floer homology of cotangent bundles and noncontractible periodic orbits

Abstract: Floer homology of the cotangent bundle of a closed Riemannian manifold M is naturally isomorphic to the homology of the loop space. An idea of proof is to relate the Floer equations in the cotangent bundle to the heat flow on the loop space. As an application we prove, given a nontrivial free homotopy class of loops in the open unit disk cotangent bundle, existence of a 1-periodic orbit representing the class, for every compactly supported time-dependent Hamiltonian which is sufficiently large over the zero section. Further applications concern a dense existence theorem and the Weinstein conjecture, both including multiplicities.

Continuity of the Green functions and the heat kernels for Schrödinger operators on Riemannian manifolds

Abstract: The Schrödinger operator $H=-\Delta+V$, where $-\Delta$ is the Laplace--Beltrami operator on a complete connected Riemannian manifold $X$ of dimension $\nu$ is considered. We suppose that $X$ is a manifold of bounded geometry and $V$ satisfies the conditions: (1) $V_+\in L^p_{\rm loc}(X)$, where $p=2$ if $\nu\le3$ and $p>\nu/2$ if $\nu\ge4$; (2) $V_-\in L^p(X)+L^\infty(X)$, where $p=2$ if $\nu\le2$ and $p>\nu-1$ if $\nu\ge3$. The continuity of the Green function of $H$ outside the diagonal and the continuity of the heat kernel for $H$ are proven. The estimates of norms of the propagator and of the resolvent in the $L^p$-scale are given. The continuity of the eigenfunctions and of the kernels for the spectral projections is discussed. The results are obtained jointly with J. Brüning and K. Pankrashkin.

Compactification of some complete Kähler manifolds with negative curvature and a strongly pseudoconvex end.

Abstract: Siu and Yau gave a differential-geometric proof of the compactification of arithmetic quotients of rank one. Our aim is to generalize their result to the case when the manifold has a strongly pseudoconvex end. Let $M$ be a connected complex manifold with compact strongly pseudoconvex boundary and of complex dimension greater than two. Assume that the interior of $M$ is endowed with a complete Kähler metric with pinched negative curvature, such that away from a neighborhood of the boundary, the volume of $M$ is finite. Then (1) the boundary of $M$ is embeddedable in some complex euclidian space and (2) $M$ can be compactified to a strongly pseudoconvex domain in a projective variety by adding an exceptional analytic set. As a consequence we extend to dimension two a theorem of Napier-Ramachandran (first announced by Burns) on quotients of the unit ball having a strongly pseudoconvex end. The talk is based on joint work with T. C. Dinh (arXiv:math.CV/0210485) and N. Yeganefar (arXiv:math.CV/0403044).

Small time heat expansion for analytic hypersurfaces with isolated singularities

We present the small time heat expansion of the Laplace-Beltrami on a hypersurface with an isolated singularity. The expansion of the distributional trace has the form tr(\chi e^{-t\Delta})\sim t^{-n/2}\sum_ju_j[\chi]t^j+ \sum_{k,j,l} v_{kj}[\chi]t^{\frac{\alpha_k}{2}-\frac12}}t^{j/2}\log ^lt where the sum over k runs over the faces of the Newton diagramm, \alpha_l are rational numbers (multiplicities of the singularity, j runs over the positive integers and the l is up to a finite positive integer. The technique is the ``irregular singularity technique'' supplemented by estimates of Hardy type at semianalytic sets. The latter roughly speaking matsch the irregular scaling of the radial variable to the variables on the link of the singularity.

Supersymmetries and Hitchin's variational principle in dimensions 7 and 8, Part I

In his recent work N.Hitchin introduced a highly original approach to special geometries in low dimensions. The starting point is the notion of a stable form which gives rise to a certain variational problem. The critical points thereof lead to complex and symplectic structures on one hand and to Riemannian structures on the other. In our talk, we shall deal with the latter but our approach to these structures is motivated by one of the most interesting features of M-theory, namely the existence of supersymmetries. From a mathematical point of view this is an identification of vectors with spinors and thus forces the structure group to reduce from SO(n) to a special subgroup. Moreover, physics gives also natural integrability conditions for these geometries (e.g. the Rarita-Schwinger or Killing equation). The aim of these talks then is to show that this approach is equivalent to the variational principle.