Research Group Differential Geometry

Secretariat
Kollegiengebäude Mathematik (20.30)
Room 1.003
Ute Peters

Institut für Algebra und Geometrie
Englerstr. 2
D-76131 Karlsruhe

Office hours:
Mo-Fr 09:00-15:00
Für Studierende:
Mo-Fr 09:15-11:15

Tel.: +49 721 608 43943

Fax.: +49 721 608 46909

# AG Differentialgeometrie (Winter Semester 2017/18)

 Lecturer: Prof. Dr. Wilderich Tuschmann (speaker of the institute) Seminar (0126600) 2

 Seminar: Wednesday 11:30-13:00 SR 2.59
 Lecturer Prof. Dr. Wilderich Tuschmann (speaker of the institute) Office hours: by appointment - please contact me by email; Room 1.002 Kollegiengebäude Mathematik (20.30) Email: tuschmann@kit.edu

Talks

25.10.2017

Huihong Jiang (Shanghai Jiao Tong University)

Bounded topology of complete manifolds with nonnegative Ricci curvature

A manifold is said to be of finite topological type if it is homeomorphic to the interior of a compact manifold with boundary. In this talk, I will give a brief introduction to the main results of manifolds with nonnegative Ricci curvature about the finite topological type. This includes some finite results under certain conditions of sectional curvature decay and diameter growth (resp. volume growth), and some counterexamples of infinite topology with positive Ricci curvature.

8.11.2017

Jan-Bernhard Kordaß (KIT)

Surgery stable curvature conditions I: Introduction

In an effort to extend a well-known result by V. Chernysh and M. Walsh, we explore the notion of a surgery stable curvature condition as suggested by the work of S. Hoelzel. We will begin with the introduction of fundamental concepts and then give a broad overview of the steps towards a proof, that will be undertaken in greater detail in subsequent talks.

15.11.2017

Jan-Bernhard Kordaß (KIT)

Surgery stable curvature conditions II: Deformations

In an effort to extend a well-known result by V. Chernysh and M. Walsh, we explore the notion of a surgery stable curvature condition as suggested by the work of S. Hoelzel. In this second talk, we will sketch the construction of a deformation map, which allows to continuously alter a riemannian metric to a certain prescribed one in a small neighbourhood of an embedded submanifold, while curvature conditions are controlled.

27.11.2017

Michael Wiemeler (Universität Münster)
SR 2.59, 2 p.m.

Non-negatively curved GKM manifolds

The classification of non-negatively curved manifolds is a long
standing open problem in Riemannian Geometry. In this talk we will
study this problem for a special class of manifolds with torus actions,
so-called GKM manifolds. We compute the cohomology rings of those GKM
manifolds which admit an invariant metric of non-negative curvature.
This is joint work in progress with Oliver Goertsches.

29.11.2017

Masoumeh Zarei (Peking University)

Equivariant classification of cohomogeneity one Alexandrov spaces in low dimensions

In this talk, I will give an equivariant classification of cohomogeneity one Alexandrov spaces in dimensions 5, 6 and 7. As a result, we show that all orbifolds appeared in the classification are equivariantly homeomorphic to a smooth good orbifold of cohomogeneity one. Fur- thermore, I will discuss a characterization of cohomogeneity one Alexandrov spaces to be topological manifolds. In particular, we prove that such spaces are homeomorphic to smooth manifolds in low dimensions. This is a joint work with Fernando Galaz-Garcia.

13.12.2017

Saskia Roos (Max Planck Institut für Mathematik, Bonn)

The Dirac operator under codimension one collapse

After giving a characterization of a collapse of codimension one we study the behavior of Dirac eigenvalues in that situation. We show that there are converging eigenvalues if and only if there is an induced spin or pin structure on the limit space. In addition, we determine the limit operator which corresponds to the limit spectrum.

16.1.2018

GGT Seminar
SR 1.067, 13:30 Uhr

Boris Vertman (Universität Oldenburg)

Stability of Ricci flow on singular spaces

We discuss recent results on the
Ricci flow for spaces with incomplete edge
singularities. In the special case of isolated cones
we establish stability of the flow near Ricci flat metrics.

17.1.2018

Martin Kerin (Universität Münster)

Non-negative curvature on exotic spheres

Since their discovery, there has been much interest in the question of precisely which exotic spheres admit a metric with non-negative sectional curvature. In dimension 7, Gromoll and Meyer found the first such example. It was subsequently shown by Grove and Ziller that all of the Milnor spheres admit non-negative curvature. In this talk, it will be demonstrated that the remaining exotic 7-spheres also admit non-negative curvature. This is joint work with K. Shankar and S. Goette.

24.1.2018

Emilio Lauret (Universidad Nacional de Córdoba)

Spectral uniqueness of bi-invariant metrics on symplectic Lie groups

Two compact Riemannian manifolds are called isospectral if their associated Laplace-Beltrami operators have the same spectra. There exist in the literature a considerable amount of pairs and families of non-isometric isospectral Riemannian manifolds. However, it is expected that Riemannian manifolds with very nice geometric attributes are spectrally distinguishable, that is, isospectrality implies isometry for them. This talk concerns the case of Riemannian symmetric spaces.

The above problem is very complicated in full generality, so it is usual to restrict the space of metrics. Gordon, Schüth, and Sutton in 2010 raised the question of whether a symmetric space given by a semisimple compact Lie group G endowed with a bi-invariant metric is spectrally distinguishable within the space of left-invariant metrics on G. A full answer was known only for 3-dimensional compact Lie groups. We will show that the question is affirmative for every symplectic group Sp(n).

30.1.2018

GGT Seminar
SR 1.067, 13:30 Uhr

Katrin Wendland (Universität Freiburg)

Refinements of the Euler characteristic, old and new

The Euler characteristic of a compact oriented manifold M is a classical invariant which by the Atiyah-Singer index theorem may be expressed as the index of a Dirac operator on M. If M is a complex manifold, then the Euler characteristic allows a refinement to the complex elliptic genus of M, as was shown in the late eighties by concerted efforts in mathematics and theoretical physics. In this talk, we will review the complex elliptic genus and explain how to further refine this invariant by methods that are motivated by conformal field theory.

7.2.2018

Michael Joachim (Universität Münster)

Twisted $spin^c$ bordism and twisted $K$-homology

In our talk we present a twisted analogue of a result of Hopkins and Hovey who show that the functor which associates to a space $X$ the graded abelian group $\Omega^{spin}_{*}(X) \otimes_{\Omega^{spin}_{*}} KO_{*}(pt)$ yields a geometric description of $KO_{*}(X)$. Our analogue for twisted $K$-theory also gives further inside to a Brown-Douglas approach to twisted $K$-homology. The results are joint work with Baum, Khorami and Schick.