### AG Differentialgeometrie (Winter Semester 2017/18)

- Lecturer: Prof. Dr. Wilderich Tuschmann (Institute Spokesman)
- Classes: Seminar (0126600)
- Weekly hours: 2

Schedule | ||
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Seminar: | Wednesday 11:30-13:00 | SR 2.59 |

Lecturers | ||
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Lecturer | Prof. Dr. Wilderich Tuschmann (Institute Spokesman) | |

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.