# Achievements of Research Area A

## Geometry and partial differential equations.

In two similar but independent approaches, Lott, Villani and Sturm developed a synthetic concept of Ricci bounds for metric measure spaces. The crucial properties of this so-called Curvature-Dimension condition are its stability under convergence of the underlying metric measure spaces and the fact that it implies a huge number of geometric and functional analytic inequalities with sharp constants, e.g. Brunn-Minkowski, Bishop-Gromov and Lichnerowicz estimates [A:Stu06a, A:Stu06b].  There is a beautiful and simple consequence: the space of equivalence classes of metric measure spaces which satisfy the CD-condition with given parameters and with uniformly bounded diameters is compact. The path-breaking observation of Otto was that the heat flow is not only the gradient flow for the energy in L2 but also the gradient flow of the entropy in the Wasserstein space of probability measures. Erbar in his Diploma thesis was the first to present a rigorous proof of this equivalence in the general setting of Riemannian manifolds [A:Erb10]. Rumpf et al. [A:NNRW09, A:LNR11] studied the numerical discretisation of geometric flows and of partial differential equation on evolving surfaces. Picking up natural time discretisation concepts for gradient flows, he recently developed a new class of variational time stepping schemes for the Willmore flow.
Wellposedness in scale invariant function spaces is a natural aim for many dispersive equations. Hadac, Herr and Koch [A:HHK09] prove wellposedness for the Kadomtsev-Petviashvili-II equation in a homogeneous critical Sobolev space with negative derivative index. The main ingredient are spaces of bounded p variation (which goes back to Wiener, and which are relevant in probability) and their predual, which improve Bourgain’s Fourier restriction spaces. Herr, Tataru and Tzvetkov [A:HTT] add a sophisticated decomposition and Bourgain’s estimates for oscillatory sums to prove wellposedness for the energy critical nonlinear Schrödinger equation on the three dimensional torus. Koch and Marzuola (HCM-Postdoc) [A:KM] combined these techniques with arguments from Martel and Merle to prove stability of the soliton in a critical Sobolev space, and scattering of the deviation from the soliton. In 2009 we organised the HCM workshop “Nonlinear structures arising in dispersive partial differential equations”.
De Lellis and Székelyhidi replace Scheffer’s and Shnirelman’s ad hoc arguments for nonuniqueness for the Euler equation by placing the question into the context of convex integration [A:DLS09]. By this they obtain a more conceptual and much more flexible proof. A line of research by Koch and Tataru culminated in a proof of strong unique continuation for parabolic equations with Lp conditions on the coefficients of the lower order terms [A:KT09]. Younger members made progress in different directions: Julie Rowlett (HCM-Postdoc) and Thalia Jeffres [A:JR10] have solved a variation of the classical Yamabe problem in the setting of conical manifolds. Semmelmann and Weingart [A:SW10] studied Weitzenböck formulas. Together with Research Area B we organised a winter school on analysis in February 2010. Thiele and Volberg received Humboldt Research Awards in 2010 resp. 2011. Their research is related to the analysis in Research Area A. There were a number of events for younger mathematicians: Tian gave the Klein Lecture 2008 on “Geometry and analysis of 4-manifolds”. There were student workshops on “Minimal surfaces and harmonic maps” and on “Kähler geometry” in 2008 and 2009.
In Research Area A we study the geometry of metric measure spaces, which is strongly related to the probabilistic view of metric measure spaces in Research Area G. The geometric and harmonic analysis side of PDEs complements the applied analysis approach to PDEs and the calculus of variations in Research Area B.

## Spectrum and index of geometric differential operators.

There is an intricate relation between geometry and the spectrum of Laplace and Dirac operators, connected with key notions like the Atiyah-Singer- and the Atiyah-Patodi-Singer-index theorem, analytic and Reidemeister torsion, scattering and automorphic forms. In the focus of current research are special classes of non-compact Riemannian manifolds such as manifolds with cusps and corners, locally symmetric spaces of finite volume, and moduli spaces of Riemann surfaces.
On complete manifolds, the structure of the continuous spectrum plays a crucial role in the analysis of geometric differential operators, such as Dirac-type operators. In [A:MS07] W. Müller and Salomonsen developed general methods to deal with scattering for Laplacians under short range perturbations of the metric on complete manifolds with bounded curvature. The important issue is that no assumption about the injectivity radius is made anymore so that natural classes of complete Riemannian manifolds, like locally symmetric manifolds, are included. W. Müller and Strohmaier [A:MS10] study the inverse scattering problem on manifolds with cylindrical ends. Scattering theory for p-forms on such manifolds has a direct interpretation in terms of cohomology. One of the main results is that the so-called scattering length, which is the Eisenbud-Wigner time delay at zero energy, can be estimated in terms of geometric data like volume and homological systoles.
The analytic torsion $T_X(\rho)$ of a compact Riemannian manifold X is a spectral invariant that depends on a representation $\rho$ of $\pi_1(\rho)$. It is defined as a weighted product of regularised determinants of the Laplacians on p-forms. Its combinatorial counterpart, the Reidemeister-Franz torsion $\tau_1(\rho)$, is known from homotopy theory. Vertman [A:Ver09a] has taken the initiative and revived the attempt to relate analytic torsion to Reidemeister torsion on manifolds with conical singularities. Vertman has computed the analytic torsion of a bounded generalised cone in terms of spectral invariants of the base.
The study of manifolds with cusps or corners involves a model geometry (pure cones or cusps), a perturbation of it, and the combination with the compact part. Lesch, Booß-Bavnbeck and Zhu [A:BBLZ09] study fundamental analytical properties of first-order elliptic operators on a compact manifold with boundary. They give a new construction for the invertible double which extends to general first order operators, and which yields a canonical construction of the Calderón projection. Related, but with a different purpose is the work of Ballmann, Brüning and Carron [A:BBC08] on elliptic boundary value problems for Dirac-type operators with rough coefficients. This provides a key tool to deal with noncompact manifolds with negative pinched sectional curvature and finite volume. The recent work of Ballmann-Brüning-Carron contains further applications, for example integrality results for volumes of complex hyperbolic manifolds.
We organised the HCM workshop “Partial differential equations and analysis on singular spaces” in 2008, the Conference “Noncommutative Geometric Methods in Global Analysis” together with Research Area C in 2009, the Conference “Spectral analysis on noncompact manifolds” together with Research Area D in 2010 and the “Bonner Geometrietage 2011”.