### AG Differentialgeometrie (Winter Semester 2019/20)

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

Schedule | ||
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Seminar: | Thursday 11:30-13:00 | SR 2.058 |

Lecturers | ||
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Lecturer | Prof. Dr. Wilderich Tuschmann | |

Office hours: by appointment | ||

Room 1.002 Kollegiengebäude Mathematik (20.30) | ||

Email: tuschmann@kit.edu | ||

Lecturer | Dr. Georg Frenck | |

Office hours: Wednesday, 10:00-11:30 or by appointment. | ||

Room 1.008 Kollegiengebäude Mathematik (20.30) | ||

Email: georg.frenck@kit.edu |

### Talks

** 24.10.2019**

**Lashi Bandara (Universität Potsdam)**

*Boundary value problems for general first-order elliptic differential operators*

The Bär-Ballmann framework is a comprehensive machine useful in studying elliptic boundary value problems (as well as their index theory) for first-order elliptic operators on manifolds with compact and smooth boundary. A fundamental assumption in their work is that an induced operator on the boundary can be chosen self-adjoint. Many operators, including all Dirac type operators, satisfy this requirement. In particular, this includes the Hodge-Dirac operator as well as the Atiyah-Singer Dirac operator. Recently, there has been a desire to study more general first-order elliptic operators, with the quintessential example being the Rarita-Schwinger operator on 3/2-spinors. In general dimensions, every induced boundary operator for the Rarita-Schwinger operator is non self-adjoint.

In this talk, I will present recent work with Bär where we and consider general first-order elliptic operators by dispensing with the self-adjointness requirement for induced boundary operators. The ellipticity of the operator allows us to understand the structure of the induced operator on the boundary, modulo a lower order additive perturbation, as bi-sectorial operator. We use a mixture of methods coming from pseudo-differential operator theory, bounded holomorphic functional calculus, semi-group theory as well as methods arising from the resolution of the Kato square root problem to extend the Bär-Ballman framework.

If time permits, I will also touch on the non-compact boundary case, and potential extensions of this to the L^p setting and Lipschitz boundary.

** 30.10.2019 (Wednesday, 11:30, SR 2.059)**

**Vicente Cortes (Universität Hamburg)**

*Generalized connections and integrability*

We characterize the integrability of various structures on Courant algebroids in terms of torsion-free generalized connections. The applications include generalized Kähler and generalized hyper-Kähler structures as particular examples. We do also give a spinorial characterization in the case of regular Courant algebroids. This is based on the theory of Dirac generating operators, for which we develop a new approach based on the geometric data encoding the regular Courant algebroid. This is joint work with Liana David, see arXiv:1905.01977.

** 06.11.2019 (Wednesday, 15:45, SR 3.069)**

**Ana Karla Garcia Perez (KIT)**

*Spaces and moduli spaces of flat Riemannian metrics on closed manifolds.*

We are going to introduce the relation between flat metrics and Bieberbach groups. This relation will provide us a way to describe the moduli space of flat metrics of a closed flat manifold, which is a result of Wolf. Then we will consider the case of 3-dimensional flat manifolds, where all the moduli spaces of flat metrics are contractible except for two. We will conclude with some comments on the 4-dimensional case.

** 14.11.2019**

**Rafael Dahmen (KIT)**

*Character groups of Hopf algebras as infinite-dimensional Lie groups*

An "infinite-dimensional Lie group" can be defined as an infinite-dimensional smooth manifold endowed with a group structure such that the group operations are smooth.

The unitary group of a Hilbert space or the diffeomorphism group of a compact (finite-dimensional) manifold are two important examples of such objects.

In this talk, I will give a brief introduction into the theory of infinite-dimensional Lie groups and what are interesting questions about them.

The guiding example will be the group of characters on a graded and connected Hopf algebra which has connections to theoretical physics and numerical analysis.

** 21.11.2019 (14:00, SR -1.008)**

**Georg Frenck (KIT)**

*The action of the mapping class group on spaces of metrics of positive scalar curvature*

In 1974, Nigel Hitchin showed that the orbit map associated to the action of the diffeomorphism group of the d-dimensional sphere on its space of positive scalar curvature metrics induces a nontrivial map on components provided that d=0,1 mod 8. Hence, this action can be used to detect non-isotopic psc metrics.

In this talk, I will present a rigidity result for this action. Among other applications, it implies that the orbit map for the sphere is trivial in every other dimension bigger than 6. So Hitchin’s detection result is the only possible one of this kind for high-dimensional spheres.

** 28.11.2019**

**Philipp Reiser (KIT)**

* Moduli Spaces of Riemannian Metrics with Positive Scalar Curvature on Topological Spherical Space Forms*

Let M be a spherical space form of dimension at least 5 which is not simply-connected. Then the moduli space of Riemannian metrics with positive scalar curvature on M has infinitely many path components as shown by Boris Botvinnik and Peter B. Gilkey in 1996. We will review this theorem which involves twisted spin structures, suitable bordism groups and eta invariants. We then show that it can be generalized to the class of topological spherical space forms, i.e. smooth manifolds whose universal cover is a homotopy sphere.

** 05.12.2019**

**Jonas Stelzig (LMU München)**

*The cohomology of para-Kähler manifolds looks nothing like that of a Kähler manifold.*

A bifoliation is a smooth manifold M together with two integrable distributions TF,TG which have complementary dimension and span the tangent space at each point. Given such a structure, one obtains two spectral sequences converging to the (real) de Rham cohomology of M and two induced filtrations on the de Rham cohomology. Formally, this situation is very similar to the case of complex de Rham cohomology of complex manifolds. In the latter case, the situation is particularly nice when the manifold admits a so-called Kähler metric: Both spectral sequences degenerate and the filtrations split. The Kähler condition has a very natural analogue in the bifoliated world, called a para-Kähler or Künneth structure, which begs the question wether similar properties to those in the bifoliated case hold. I will report on work in progress with L. Garcia and D. Kotschick, where we construct a sequence of examples showing that even in such restricted cases all nice properties one might wish for fail rather drastically.

** 12.12.2019 **

**Xiaoyang Chen (Shanghai University)**

*Morse-Novikov cohomology of almost non-negatively curved manifolds*

** 19.12.2019**

**David Degen (KIT)**

*Moduli Spaces of Ricci-Flat Metrics on K3 Surfaces*

K3 surfaces are complex manifolds which are particularly nice for studying Ricci-flat metrics, for example one has a good description of the moduli space of Ricci-flat metrics (including orbifold metrics). In this talk I will show that the first Betti number of this moduli space vanishes.

If time permits, I will also talk about convergence of Ricci-flat metrics on K3 surfaces.

** 09.01.2020**

**Xiaolei Wu (Universität Bielefeld)**

*Riemannian foliation with exotic tori as leaves*

Fernando Galaz-Garcia and Marco Radeschi asked whether there exists a singular Riemanian foliation on a simply connected manifold whose leaves are exotic tori. We discuss a construction of smooth fiber bundles such that the fibers are exotic tori and the total space has finite abelian fundamental group. This gives examples of a Riemannian foliation on a closed manifold whose leaves are exotic tori and whose total space has finite abelian fundamental group. This is a joint work with F. Thomas Farrell.

**30.01.2020**

**Martin Günther (KIT)**

*The fundamental (semi-)category of a Lorentzian manifold*

In this talk, we define and investigate elementary properties of the "fundamental category/semicategory" of a Lorentzian spacetime, whose morphisms are homotopy classes of causal or chronological paths, respectively. The definitions are inspired by Marco Grandis's directed algebraic topology.

The obtained results suggest that these structures could be useful in the investigation of Lorentzian manifolds with weak or no causality conditions, since they encode local and global features of the causal structure and the manifold topology.