# Program

Friday | (Physics Building, Room 2.73) | Saturday | (Biology Building, Room 0.110) |
---|---|---|---|

9:15 - 10:00 | Michael Wiemeler | ||

10:15 - 11:15 | Jost-Hinrich Eschenburg | ||

11:30 - 12:15 | Mark Hamilton | ||

13:00 - 13:45 | Peter Jossen | 12:15 - 13:45 | Lunch |

14:00 - 14:30 | Peter Quast | 13:45 - 14:30 | Tillmann Jentsch |

14:45 - 15:15 | Johannes Riesterer | 14:45 - 15:30 | Malte Röer |

15:45 - 16:15 | Fernando Galaz-Garcia | 15:45 - 16:15 | Alexander Engel |

16:30 - 16:45 | Micha Wasem | 16:30 - 17:15 | Gabriel Ben Simon |

17:00 - 18:00 | Michael Joachim | 17:30 - 18:00 | Mircea Petrache |

19:00 - … | Conference dinner at the Gemelli |

## Michael Joachim (Münster)

### Exotic spheres and curvature

In this talk we will survey some classical results concerning curvature and special metrics on homotopy spheres. In particular we will address the relevance of the so-called plumbing construction for applications in this geometric context.

## Jost-Hinrich Eschenburg (Augsburg)

### Riemannian geometry and linear algebra

Bernhard Riemann in his famous inauguration lecture said about higher-dimensional manifolds: "More frequent occasions for the creation and development of these notions occur first in higher mathematics". Probably he was thinking at certain algebraic manifolds, like projective spaces and Grassmannians. These belong to an interesting subclass, symmetric spaces, which are closely related to the most basic "higher mathematics", to linear algebra.

We show three instances where this relationship becomes important:

- Normal forms of certain matrices and polarity,
- Bott periodicity,
- Exceptional spaces and the octonions.

The linear algebra happens over the real division algebras, the real, complex, quaternionic and octonionic numbers. The octonions will appear at all three instances; their relation to symmetric spaces is least understood since a "linear algebra" over the octonions does not (yet) exist. This is ongoing work.

## Peter Jossen (Paris)

### Around the Hodge conjecture for complex tori

The cohomology of every smooth compact Kähler Manifold carries a Hodge-Structure, and one can formulate the Hodge conjecture (which is for complex *algebraic* varieties) verbatim for Kähler Manifolds.

S. Zucker (1977) and C. Voisin (2002) have shown that this more general statement is wrong. Their counterexamples are certain complex tori.

I plan to start with stating the Hodge conjecture in elementary terms for complex tori, and recall some classical positive results. Then I shall explain what goes wrong in the examples of Zucker and Voisin. Whatever time remains I will use to say something about Weil-classes (these are cohomology classes which play an essential role in Voisin's example), Weil-motives and their fundamental group.

## Peter Quast (Augsburg)

### Convexity of reflective submanifolds in symmetric $R$-spaces

We show that reflective submanifolds in symmetric $R$-spaces are geodesically convex and we give counterexamples in larger categories of compact symmetric spaces (joint work with M.S. Tanaka).

## Johannes Riesterer (Karlsruhe)

### Automorphism groups of Lie algebras

We look at the automorphism group of Lie algebras from Sullivan's infinitesimal viewpoint.

## Fernando Galaz-Garcia (Münster)

### Simply connected 5-manifolds with nonnegative sectional curvature and almost maximal symmetry rank

I will discuss the topological classification of simply connected 5-manifolds with nonnegative curvature and an isometric action of the 2-torus.

## Micha Wasem (Zürich)

### Isometric embeddings with prescribed boundary data

We will discuss certain aspects of a Cauchy type problem for isometries at high (resp. low regularity). We present some results and an open problem.

## Michael Wiemeler (Bonn)

### Torus actions and positive scalar curvature

We discuss a construction of invariant metrics of positive scalar curvature on certain manifolds with torus actions and some of its consequences.

## Mark Hamilton (Stuttgart)

### Representing homology classes by symplectic surfaces

We will explain how branched coverings can be used to show that certain homology classes in symplectic 4-manifolds cannot be represented by embedded, possibly disconnected, symplectic surfaces.

## Tillmann Jentsch (Stuttgart)

### On parallel submanifolds of symmetric spaces

A submanifold of a Riemannian symmetric space is called parallel if its second fundamental form is parallel. We classify parallel submanifolds of the Grassmannian $G^+_2(\mathbb{R}^{n+2})$ which parameterizes the oriented 2-planes of the Euclidean space $\mathbb{R}^{n+2}$. Our main result states that every complete parallel submanifold of $G^+_2(\mathbb{R}^{n+2})$, which is not a curve, is contained in some totally geodesic submanifold as a symmetric submanifold. This result holds also if the ambient space is the non-compact dual of $G^+_2(\mathbb{R}^{n+2})$.

## Malte Röer (Regensburg)

### On the bordism-invariance of the $KO$-index

It is well known that the $KO$-index of Dirac operators on spin manifolds is invariant under spin cobordism. We give a new proof of this using geometrically constructed homotopies of Dirac operators.

## Alexander Engel (Augsburg)

### Isospectral Alexandrov Spaces

Two Alexandrov spaces are called *isospectral* if the eigenvalues of their Laplace operator coincide, counted with multiplicities. We construct the first examples of Alexandrov spaces which are isospectral, but not isometric.

This is joint work with Martin Weilandt (ufsc brasil).

## Gabriel Ben Simon (Zürich)

### Causal Representations of Surface Groups

An important sub-class of Riemannian Manifolds is the Hermitian symmetric spaces (this means symmetric spaces for which the Riemannian metric comes from a hermitian structure on the space).

Given a surface group of a closed, say, surface of genus at least 2, a well studied moduli space is the space of homomorphisms of the surface group into the (Lie) group of isometries of a given Hermitian symmetric space. In particular sub-spaces of these moduli spaces are used to generalize the notion of a Teichmuller space. We will present two new invariants of these moduli sub-spaces for the case of the Poincare disc, a basic Hermitian symmetric space ( the group of isometries in this case is $PSL_2 (\mathbb{R})$) and thus of Teichmuller spaces. These invariants are used to generalize Teichmuller spaces while replacing $PSL_2(\mathbb{R})$ with general Hermitian Lie groups. We will explain why this generalization is worthwhile and present a right candidate for a generalization.

This project is based on works by Burger-Iozzi-Wienhard; Ben Simon-Hartnick; and Ben Simon-Burger-Hartnick-Iozzi-Wienhard.

## Mircea Petrache (Zürich)

### Constructing nontrivial $U(1)$-bundles by variational methods

We will show how to construct topologically non-trivial $U(1)$-bundles with singularities over a 3-dimensional manifold with (possibly empty) boundary, by minimizing the $L^p$ energy of their curvatures. This requires to define, analogously to Sobolev functions, a notion of "singular bundles" such that the topological invariant, i. e. the first Chern class, is preserved under sequential weak convergence.

After showing the existence of minimizers in this wider class, we will describe the regularity theory, which shows that minimizers coincide with "classical bundles" defined (and smooth) outside a finite number of points, around each of which a nontrivial Chern class is realized. The interest of this construction is that both the number and position of the singular points (or "topological charges") are optimized via the energy minimization requirement and therefore they gain an intrinsic geometric relevance.

After discussing the relation with harmonic maps and Yang-Mills theory, we will describe different constraints which can be applied (e.g. one can prescribe a "boundary value" for the bundles, as for minimal surfaces) and their influence on the number and positions of charges.