KIT - Fakultät für Mathematik

In this talk I want to give an overview over some old and new results about the (smooth) Yamabe invariant in joint work with M. Dahl and E. Humbert. We will see for example that the Yamabe invariant of a simply connected compact spin manifold of dimension 5 is between 45 and 79. Similar estimates hold for 2-connected compact spin manifolds with vanishing index. more...

Oliver Baues, Karlsruhe
Fibrations of aspherical Riemannian manifolds

Every compact aspherical Riemannian manifold admits a canonical series of fibrations which are Riemannian orbibundles with infrasolv-fibers. Each step of the resulting tower of maps arises from the action of the continuous part of the isometry group on the universal cover. The length of the tower and the geometry of its base measure the degree of continuous symmetry of an aspherical Riemannian manifold. The symmetry is called large if the base is locally symmetric. We show that closed aspherical manifolds with exotic smooth structure do not support a metric with large symmetry.

Michelle Bucher-Karlsson, Genf
Euler Characteristic, Simplicial volume and the Schläfli volume formula

I will consider two topological invariants of manifolds, the classical Euler characteristic and the newer simplicial volume (introduced by Gromov in 1980), and discuss their analogies and differences. In particular, I will discuss a question of Gromov, asking if the vanishing of the simplicial volume of an aspherical manifold implies the vanishing of its Euler characteristic. Based on a generalization of the classical Schläfli volume formula, I will answer this question for a modified version of the simplicial volume, defined for immersed simplices.

Luis Guijarro, Madrid
The cut locus and balanced split loci in Riemannian manifolds

We will review recent results on the cut loci of points or hypersurfaces in a Riemannian manifold. More precisely, we characterize the cut locus as a balanced split locus; conversely, we examine what other possible balanced split loci exist for a given metric. This is joint work with Pablo Angulo-Ardoy (UAM).

Bernhard Hanke, Augsburg
Large and small group homology

For several instances of metric largeness like enlargeability or having hyperspherical universal covers, we construct non-large vector subspaces in the rational homology of finitely generated groups. The functorial properties of this construction imply that the corresponding largeness properties of closed manifolds depend only on the image of their fundamental classes under the classifying map.
This is applied to construct examples of essential manifolds whose universal covers are not hyperspherical, thus answering a question of Gromov (1986), and, more generally, essential manifolds which are not enlargeable.

Lizhen Ji, Ann Arbor
Geometry and analysis of moduli spaces of Riemann surfaces

In this talk, I will describe several results on the Riemannian geometry and spectral theory of moduli spaces of Riemann surfaces.

Matthias Kreck, Bonn
Equivariant Poincaré Duality

Ordinary equivariant (co)homology is usually defined via the Borel construction. Poincaré duality fails as much as it can. Using stratifolds one can define a new cohomology theory which is Poincaré dual to ordinary equivariant homology and similarly a new homology theory which is Poincaré dual to ordinary cohomology. In a similar spirit one can define Bredon-type (co)homology theories which might be useful for equivariant characteristic classes. I hope to demonstrate the use of these theories in some examples.

Alexander Lytchak, Münster
Spherical Buildings and Polar Foliations

Spherical buildings are rigid combinatorial objects introduced by Tits as an important tool in the study of algebraic groups. Due to their rigid nature, it does not happen often that a building appears in an a priori non-algebraic situation, and every time it happens, the consequences are very strong. In my talk, I will discuss a few applications of spherical buildings in Riemannian geometry.

Hugo Parlier, Fribourg
The geometry of surfaces of large genus

What can a surface of large genus look like? What does a typical or random surface look like? In certain contexts, the talk will be about studying the behavior of curves on surfaces to understand the geometry of surfaces and their related moduli spaces. In the opposite direction, I'll discuss some work with Guth and Young about what we can learn about random surfaces by studying their moduli spaces. Various parts of the talk are joint with W. Cavendish, with F. Balacheff and S. Sabourau, and with L. Guth and R. Young.

Uwe Semmelmann, Stuttgart
Almost complex structures on quaternion Kähler manifolds and inner symmetric spaces

It is a classical result of Hirzebruch that the quaternion projective spaces do not admit any almost complex structure. In my talk I will show how the Atiyah-Singer index theorem can be used to give a short alternative proof, which applies to a much larger class of manifolds.
With a similar idea it is possible to decide which inner symmetric spaces admit almost complex structures.

Richard Weidmann, Kiel
Makanin-Razborov diagrams for hyperbolic groups

For an arbitrary finitely presented group $G$ and a free group $F$ , the set $Hom(G,F)$ was described by Razborov relying on earlier work of Makanin. This description was refined by Kharlampovich and Myasnikov and independently by Sela. Sela further generalized his approach to describe $Hom(G,\Gamma)$ for an arbitrary word-hyperbolic group $\Gamma$ . A full understanding of $Hom(G,\Gamma)$ is one of the basic building blocks for understanding the elementary theory of $\Gamma$ . It is the aim of this talk to outline Sela's approach to Makanin-Razborov diagrams; incremental contributions by the speaker and Reinfeldt are discussed if time permits.

Gabriela Weitze-Schmithüsen, Karlsruhe
Origamis in Teichmüller space
Teichmüller disks are isometric and holomorphic images of the Poincaré upper half-plane in Teichmüller space (endowed with the Teichmüller metric). There is a special interest on those, whose image in moduli space is an algebraic complex curve, in this case called Teichmüller curve. Teichmüller disks are obtained by a handy construction using translation surfaces. We study examples called square-tiled surfaces or origamis that arise from gluing finitely many copies of the Euclidean unit square along their edges. These surfaces can be represented by simple combinatorial data. We explain how the corresponding Teichmüller disks are also naturally embedded into Culler-Vogtmann outer space $CV_n$ , the classification space of marked metric graphs of genus n, and give applications of this.

Burkhard Wilking, Münster
Applications of the Margulis Lemma

We (joint work with V. Kapovitch) establish a Margulis Lemma for manifolds with lower Ricci curvature bound. In this talk I will focus on some of the applications of the Margulis Lemma rather than the proof of it.