Kirwan polytopes and representations
Monday, 7.5.12, 16:15-17:15, Raum 404, Eckerstr. 1
Kähler-Einstein metrics with edges
Monday, 14.5.12, 16:15-17:15, Raum 404, Eckerstr. 1
I will discuss the geometric problem of finding KE metrics which are bent along a divisor, and the equivalent analytic problem of solving the associated singular Monge-Ampere equation. There are several interesting applications for these metrics. Furthermore, the classical Aubin-Yau estimates do not work in this setting and a new route to the solvability of this equation must be found. This is joint work with Jeffres and Rubinstein.
Kozykel für charakteristische Klassen in der glatten Deligne-Kohomologie
Monday, 21.5.12, 16:15-17:15, Raum 404, Eckerstr. 1
Aufbauend auf den Ergebnissen meines letzten Vortrags werde ich (nach einer kurzen Wiederholung) die Unabhängigkeit von getroffenen Wahlen und die Natürlichkeit der Konstruktion untersuchen.
Monday, 11.6.12, 16:15-17:15, Raum 404, Eckerstr. 1
Scattering and length rigidity on some Riemannian manifolds with trapped geodesics
Monday, 18.6.12, 16:15-17:15, Raum 404, Eckerstr. 1
We discuss how to show that the flat solid torus is scattering rigid.\nWe will consider compact Riemannian manifolds M with boundary N. We\nlet IN be the unit vectors to M whose base point is on N and point\ninwards towards M. Similarly we define OUT. The scattering data\n(loosely speaking) of a Riemannian manifold with boundary is map from\nIN to OUT which assigns to each unit vector V of IN a unit vector\nW in OUT. W will be the tangent vector to the geodesic determined by\nV when that geodesic first hits the boundary N again. This may not be\ndefined for all V since the geodesic might be trapped (i.e. never hits\nthe boundary again). A manifold is said to be scattering rigid if any\nother Riemannian manifold Q with boundary isometric to N and with the\nsame scattering data must be isometric to M.\nIn this talk we will discuss the scattering rigidity problem and\nrelated inverse problems. There are a number of manifolds that are\nknown to be scattering rigid and there are examples that are not\nscattering rigid. All of the known examples of non-rigidity have\ntrapped geodesics in them.\nIn particular, we will see that the flat solid torus is scattering\nrigid. This is the first scattering rigidity result for a manifold\nthat has a trapped geodesic. The main issue is to show that the unit\nvectors tangent to trapped geodesics in any such Q have measure 0 in\nthe unit tangent bundle of Q. We will also consider scattering\nrigidity of a number of two dimensional manifolds (joint work with\nPilar Herreros) which have trapped geodesics..\n\n
Farey-Graph und Zopfgruppe
Monday, 25.6.12, 16:15-17:15, Raum 404, Eckerstr. 1
Area growth and rigidity of surfaces without conjugate points
Monday, 25.6.12, 16:15-17:15, Raum 404, Eckerstr. 1
A complete Riemannian manifold has no conjugate points if each of its geodesics is homotopically minimizing, i.e. if each lift of each geodesic to the universal Riemannian covering is minimizing. A theorem of E. Hopf from 1948 states that 2-tori with no conjugate points are flat. We prove flatness in case of the plane and the cylinder under optimal conditions on the area growth. The area growth of a surface is defined as the limit inferior, as r tends to infinity, of the quotient of the area of the metric r-ball about an arbitrarily fixed point and the area of a metric r-ball in the Euclidian plane. We prove that a complete plane with no conjugate points has area growth greater than or equal to one, and that equality holds only in the Euclidian case. The results were obtained in joint work with Victor Bangert.
Totalkrümmung immersierter Laminationen mit transversalem Maß
Monday, 9.7.12, 16:15-17:15, Raum 404, Eckerstr. 1
Die infinitesimal-äquivariante Eta-Invariante
Monday, 16.7.12, 16:15-17:15, Raum 404, Eckerstr. 1
Eta-Formen für Familien mit integrabler horizontaler Distribution
Monday, 23.7.12, 16:15-17:15, Raum 404, Eckerstr. 1
About a de Rham complex describing intersection space cohomology in a non-isolated singularity case
Wednesday, 25.7.12, 16:15-17:15, Hörsaal Weismann-Haus Albertst. 21
For manifolds Poincaré duality is one of the most important properties\nof singular (co)homology theory. However proceeding to singular\nspaces, in general ordinary singular (co)homology does not satisfy \nPoincaré duality no more. But there are several generalized \n(co)homology theories\nfor pseudomanifolds that satisfy Poincaré duality. One of those\ntheories is M. Banagl's (co)homology theory of Intersection Spaces.\nIn [Ban11] M. Banagl derived an alternate description of Intersection\nSpace cohomology of a stratified pseudomanifold X, in cases where one\nhas a singular stratum with flat link bundle endowed with a Riemannian\nmetric such that the structure group of the bundle is contained in the\nisometries of the link. For that purpose he makes use of a certain\nsubcomplex of the complex of differential forms on M, the non-singular\npart of X. In the isolated-singularity case the existence of an\nisomorphism between the two descriptions was shown.\nWe want to generalize this De Rham isomorphism to the non-isolated \nsingularity case where we have a trivial link bundle. We therefore \nmake use of the Künneth-theorem.\n