Goals of Research Area A
Geometry and partial differential equations.
We want to improve our understanding of the local structure of spaces satisfying the curvature-dimension condition. Do these spaces have integer Hausdorff dimension? Is there a (weak version of the) splitting theorem, at least on the level of measures? Can we characterize the metric measure spaces for which we obtain equality in functional analytic inequalities? Is there a smooth analysis on metric measure spaces satisfying the curvature-dimension condition? This condition should allow one to deduce Lipschitz estimates for heat kernels and harmonic functions in contrast to better understood pointwise estimates based on volume doubling and scale invariant Poincaré inequalities on balls. We aim for results in the spirit of the gradient estimates of Bakry-Emery and the differential Harnack inequality of Li-Yau.
Geometric flows will be studied from a number of related perspectives. We aim to study evolutions of metric measure spaces. The starting point will be the investigation of the Ricci flow on Riemannian manifolds with singularities. We will continue with curvature flows for (suitable classes of) Finsler and Alexandrov spaces. On the PDE side rough solutions to nonlinear parabolic equations, including the Ricci-de-Turck flow with initial data with metrics, will be studied. Furthermore, we aim at numerical discretizations of higher order flows in Finsler geometry which rely on a direct discretization of the geometry and underlying variational principles instead of discretizing the resulting (higher order) differential operators.
The soliton resolution conjecture expresses the expectation that solutions to dispersive equations decompose into a number of solitons, plus a scattering part. This conjecture puts a strong emphasis on an understanding of solutions in a neighborhood of the soliton set. We want to analyze the dynamics in a neighborhood of solitons, blow-up in the super-critical regime, large time behavior and the relation to more complete models.
The Lagrangian isotopy problem will be approached using minimal discs. A first step consists in the construction of fillings for Lagrangian tori T in by holomorphic discs. Secondly, we plan to extend the theory to Kähler surfaces of non-positive sectional curvature. Our main goal will be to classify knots which arise at Reeb orbits on boundaries of compact convex domains in such manifolds.
Spectrum and index of geometric differential operators.
We want to study spectral invariants of Laplace operators on noncompact and singular manifolds by analytic and topological methods. Recently W. Müller has studied the asymptotic behavior of the analytic torsion for hyperbolic 3-manifolds with respect to the symmetric powers of a given representation. This has interesting applications to the cohomology of arithmetic groups. The goal is to extend this to locally symmetric spaces of higher rank and to the finite volume case. In three dimensions this is expected to have important applications to knot theory. Also of interest is a gluing theorem for the analytic torsion in a fairly general singular setting, which will reduce the problem of comparing the analytic torsion and its combinatorial counterpart to the model of the singularity. For complete manifolds which arise as universal covering of a closed Riemannian manifold, spectral invariants (which are called L2–invariants then) can often be studied by topological methods. We will work on the Atiyah Conjecture which predicts that the L2-Betti numbers are integers if the fundamental group is torsionfree. A prototype is Lück’s theorem predicting that the L2-Betti number is the limit of the normalized L2-Betti numbers over Q for a tower of finite coverings. The case where Q is replaced by a field of prime characteristic will be studied. A harder and very interesting problem is whether one can replace the Betti numbers by the minimal numbers of generators, the order of the torsion subgroup of the homology groups or by torsion invariants (Bergeron-Venkatesh conjecture).
We will extend the research on boundary value problems and spectral theory of generalized Dirac operators to subelliptic problems as in complex analysis and to manifolds with more general ends, e.g. those which occur for manifolds of Q-Rank-1.
Harmonic analysis on locally symmetric spaces of finite volume is closely related to the theory of automorphic forms and has important implications for number theory. The fine structure of the continuous spectrum is intimately connected with basic properties of the discrete spectrum and cuspidal automorphic forms. We want to develop methods of geometric scattering theory to deal with the continuous spectrum of higher rank locally symmetric spaces of finite volume. There are close similarities with spectral theory of Hamiltonians associated to N-body interactions, and the goal is to adapt the methods from this theory to the geometric setting. The moduli space of Riemann surfaces of a fixed genus with a finite number of punctures shares many similarities with locally symmetric spaces of finite volume. One of the goals is to develop spectral theory for moduli spaces similar to the locally symmetric case.