Abstracts

Werner Ballmann, Universität Bonn
Small eigenvalues of the Laplacian on surfaces
I will discuss my recent joint work with Henrik Matthiesen and Sugata Mondal, where we prove a generalized version, and by now also in the noncompact case, of the conjecture $\lambda_{2g-2}>1/4$ of Buser and Schmutz. (The original proof of the BS-conjecture is by Otal and Rosas, we extend their work.)


Christian Bär, Universität Potsdam
An index theorem for compact Lorentzian manifolds with boundary
We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. The index is also shown to equal that of a certain operator constructed from the evolution operator and a spectral projection on the boundary. In case the metric is of product type near the boundary a Feynman parametrix is constructed.

This is the first index theorem for Lorentzian manifolds and, from an analytic perspective, the methods to obtain it are quite different from the classical Riemannian case. This is joint work with Alexander Strohmaier.


Renato Bettiol, University of Pennsylvania
Strongly positive curvature: homogeneous classification and some obstructions
Strongly positive curvature is an intermediate condition between positive-definiteness of the curvature operator and positive sectional curvature ($\sec>0$), defined in terms of adding a 4-form to the curvature operator to make it positive-definite. It stems from the work of Thorpe in the 1970s, but has also been implicitly studied by others. The classification of simply-connected homogeneous spaces with strongly positive curvature has been recently completed, propelled by the discovery that Riemannian submersions preserve this curvature condition. After establishing that most examples with $\sec>0$ actually satisfy this stronger condition, much of the current interest is in restricting the topology of such manifolds. In this talk, I will describe these recent developments and report on work in progress regarding applications of the Bochner technique to find topological obstructions to strongly positive curvature. This is joint work with R. Mendes (WWU Münster).


Anand Dessai and Martin Herrmann, Université de Fribourg and Karlsruher Institut für Technologie
Curvature, cohomogeneity less than or equal to one, and complex cohomology
The talk consists of two parts, both related to a question of Karsten Grove asking whether an upper bound on the diameter and a lower bound on the curvature of a simply connected n-manifold imply finiteness of the possible rational homotopy types.

In the first part we construct several infinite families of nonnegatively curved manifolds of low cohomogeneity and small dimension which can be distinguished by their cohomology rings. In particular, we exhibit an infinite family of eight-dimensional cohomogeneity-one manifolds of nonnegative curvature with pairwise non-isomorphic complex cohomology rings.

In the second part we construct new 13- and 22-dimensional families of counterexamples to Grove's question among simply connected, nonnegatively curved, homogeneous spaces with pairwise non-isomorphic complex cohomology rings. The 22-dimensional family in addition admits homogeneous metrics with almost nonnegative curvature operator.


Fernando Galaz-Garcia, Karlsruher Institut für Technologie
Cohomogeneity-one topological manifolds
Cohomogeneity one topological manifolds were introduced by Mostert in 1957. In this talk I will discuss the structure of these spaces and show that, in contrast to what was originally claimed by Mostert, not every cohomogeneity one topological manifold is equivariantly homeomorphic to a smooth cohomogeneity one manifold. I will then discuss the classification of closed, simply connected cohomogeneity one topological manifolds in dimensions 5, 6, and 7. In these dimensions, these manifolds are homeomorphic to smooth manifolds. This is joint work with Masoumeh Zarei.


Nadine Große, Universität Leipzig
The $L^p$ spectrum of the Dirac operator
We study the $L^p$-spectrum of the Dirac operator on complete manifolds. One of the main questions in this context is whether this spectrum depends on $p$. As a first example where $p$-independence fails we compute explicitly the $L^p$-spectrum for the hyperbolic space and its product with compact spaces. These are the first computations of the $L^p$-spectrum for the Dirac operator on noncompact manifolds. This is joint work with Bernd Ammann.


Karsten Grove, University of Notre Dame
Diameter rigidity in positive curvature: new and old
Recall that a manifold with curvature at least $1$ has diameter at most $\pi$ (Bonnet), and is isometric to the unit sphere when the diameter is maximal (Toponogov). It turns out that the base $B$ of a (nontrivial) Riemannian submersion from a manifold $M$ with curvature at least $1$ has diameter at most $\pi/2$ and, when maximal, $B$ is rigidly determined.

We will discuss this and more general rigidity results for submetries in this context (Joint work with Xiaoyang Chen). Prior joint work with Gromoll and work of Wilking on Riemannian submersions from spheres and diameter rigidity for positively curved manifolds with half maximal diameter play crucial roles and will be discussed as well.


Luis Guijarro, Universidad Autónoma de Madrid
Every point is critical in a Riemannian manifold
We show that in any compact Riemannian manifold $M$, any point $p$ is critical for the distance function from some other point $q$. This extends results of of Bárány, Itoh, Vîlcu and Zamfirescu, who proved a similar statement for Alexandrov surfaces. This is joint work with Fernando Galaz-Garcia.


Lee Kennard, University of Oklahoma
Torus actions and positive curvature in dimensions up to 24
It is a classical question to understand which smooth manifolds admit Riemannian metrics with positive sectional curvature. In the presence of torus symmetry, fundamental papers by Hsiang-Kleiner, Grove-Searle, Wilking, and others have contributed to understanding of this question. Work of Dessai provides sharp calculations of a number of topological invariants for 8-manifolds with positive curvature and torus symmetry, both in general and under additional assumptions (e.g., rationally elliptic). In joint work with Manuel Amann, we have furthered this work into dimensions up to 24. I will summarize our calculations, discuss relevant examples, and discuss what computational techniques are involved in our proofs.


Ramiro Lafuente, Universität Münster
Homogeneous Ricci flows
In this talk we will discuss the basic aspects of the Ricci flow of homogeneous manifolds, presenting some recent developments and explaining its relevance and potential applications. After that we will focus on the case of solvable Lie groups. We will show that every Einstein solvmanifold (and more generally, every solvable algebraic soliton) is stable under the flow for left-invariant metrics on the same group. Finally, we will show -in some special cases- why the flow converges to a soliton metric, and whether these are global atractors. This is based on a joint work in progress with Christoph Böhm.


Nina Lebedeva, Steklov Institute, St. Petersburg
Total curvature of geodesics on convex surfaces
We prove that the total curvature of a minimizing geodesic segment on a convex surface in the 3-dimensional Euclidean space can not be arbitrarily large. (Joint with Anton Petrunin)


Alexander Lytchak, Universität zu Köln
Isometric characterization of non-positive curvature
In the talk I would like to present the main ideas involved in the proof of the following joint result with S. Wenger:

A locally compact geodesic metric space has non-positive curvature (a CAT(0)-space) if and only if it satisfies the 2-dimensional Euclidean isoperimetric inequality, hence if and only if any closed curve of length L bounds some disc of area at most $L^2/4\pi$. The main steps in the proof are the solution of the classical Plateau problem in general spaces and the analysis of the intrinsic structure of area minimizing discs. The result also applies to non-zero upper bounds on curvature.


Andrea Mondino, ETH Zürich
Some properties of non-smooth spaces with Ricci curvature lower bounds
The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80s and was pushed by Cheeger and Colding in the '90s who investigated the structure of the spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds.

A completely new approach via optimal transportation was proposed by Lott-Villani and Sturm almost ten years ago; with this approach one can a give a precise meaning of what means for a non smooth space to have Ricci curvature bounded from below by a constant. This approach has been refined in the last years by a number of authors (let me quote the fundamental work of Ambrosio-Gigli-Savaré among others) and a number of fundamental tools have now been established (for instance the Bochner inequality, the splitting theorem, etc.), permitting to give further insights in the theory. In the seminar I will give an overview of the topic.


Alexander Nabutovsky, University of Toronto
Balanced finite presentations of the trivial group and geometry of four-dimensional manifolds
Recently Boris Lishak has constructed a sequence of finite presentations of the trivial group with just two generators and two relators such that the minimal number of relations required to demonstrate that a generator is trivial grows faster than the tower of exponentials of any fixed height of the length of the finite presentation.

I will explain this result and some of its implications to Riemannian geometry of four-dimensional manifolds. For example, for each closed four-dimensional Riemannian manifold $M$ and each sufficiently small positive $\epsilon$ the set of isometry classes of Riemannian metrics on $M$ of volume one with the injectivity radius greather than or equal to $\epsilon$ is disconnected. A similar disconnectedness result holds for sets of Riemannian structures with $\sup\vert K \vert diam^2 \leq X$ on each closed four-dimensional manifold with non-zero Euler characteristic providing that $X$ is sufficiently large. (A joint work with Boris Lishak.)


Marco Radeschi, Universität Münster
Metrics on spheres all of whose geodesics are closed
Riemannian manifolds in which every geodesic is closed, have been studied since the beginning of last century, when Zoll showed the existence of a non-round metric on the 2-sphere all of whose geodesics are closed. Among the many open problems on the subject, a conjecture of Berger states that for any simply connected manifold all of whose geodesics are closed, the geodesics must have the same length. The result was proved in the case of the 2-sphere by Grove and Gromoll. In this talk, I will show recent work with B. Wilking, where the Berger conjecture is proved for spheres of dimension >3.


Regina Rotman, University of Toronto
Quantitative homotopy theory and the lengths of geodesics on Riemannian manifolds
Let M be a closed Riemannian manifold. There are numerous results that establish the existence of various minimal objects on M, such as periodic geodesics, minimal surfaces, or geodesic nets. We will present some effective versions of these existence theorems.

For example, we will present diameter upper bounds for the lengths of three shortest simple periodic geodesics on a Riemannian 2-sphere, which can be interpreted as an effective version of the existence theorem of Lusternik and Schnirelmann. (Joint with Y. Liokumovich and A. Nabutovsky).

Finding upper bounds for the size of smallest stationary objects is closely related with construction of "optimal" homotopies. We will show that if M is a closed surface of diameter d (with or without boundary), then any simple closed curve on M that can be contracted to a point over free loops of length less than L, can be contracted over based loops of length at most 3L+2d. (Joint with G. Chambers).


Krishnan Shankar, University of Oklahoma
On semi-free actions whose orbit spaces are manifolds
Semi-free $S^1$ actions have been studied in some detail on spheres by Montgomery, Yang and Levine, among others. One interesting case of such actions is when the fixed point set has codimension 4; in this case the orbit space admits a smooth manifold structure. For instance, all exotic 7-spheres admit infinitely many such actions with quotient space $\mathbf{S}^6$; the actions are classified by the knotting of the fixed point set, $\mathbf{S}^3$. In this talk we will explore which 5-manifolds admit such actions with quotient a simply connected 4-manifold. We will show:

Suppose $M^5$ is (i) a simply connected 5-manifold admitting a semi-free $S^1$ action with (ii) codimension-4 fixed point set with $n$ components and (iii) orbit space (manifold) $M^*$ with $b_2(M^*) = k$. Then $M$ is a connected sum of $n+k-1$ copies of $\mathbf{S}^3$-bundles over $\mathbf{S}^2$. Moreover, $M$ is spin if and only if $M^*$ is spin.

Time permitting we will also talk about semi-free $S^3$ actions on 8-manifolds and a generalization of this type of action with manifold quotient. This is joint work, in progress, with John Harvey and Martin Kerin.


Anna Siffert, University of Pennsylvania
Isoparametric hypersurfaces in spheres
The problem of classifying isoparametric hypersurfaces in spheres is still not completely solved. In addition to explaining fundamental gaps in recent contributions to the case with six different principal curvatures, I present a new method for classifying isoparametric hypersurfaces of spheres with six different principal curvatures all of multiplicity one. This approach yields also gives us a potential strategy for classifying isoparametric hypersurfaces of spheres with six different principal curvatures all of multiplicity two, which is still open.


Michael Wiemeler, Universität Augsburg
Invariant metrics of positive scalar curvature on $S^1$ manifolds
In this talk we will discuss the construction of invariant metrics of positive scalar curvature on manifolds $M$ with circle actions. We will discuss two cases. First the case where there is a fixed point component of codimension two. Then there is always an invariant metric of positive scalar curvature on $M$.

The case where the fixed point set has codimension at least four is more complicated. In this case the answer to the question if there is an invariant metric of positive scalar curvature on $M$ depends on the class of $M$ in a certain equivariant bordism group. We will discuss the case where the maximal stratum of $M$ is simply connected and all normal bundles to the singular strata are complex vector bundles in more detail. In this case there is an $l\in \mathbb{N}$ such that the equivariant connected sum of $2^l$ copies of $M$ admits an invariant metric of positive scalar curvature if and only if a $\mathbb{Z}[\frac{1}{2}]$-valued bordism invariant of $M$ vanishes.


David Wraith, National University of Ireland, Maynooth
Positive Ricci curvature on highly connected manifolds
This talk concerns the existence of positive Ricci curvature metrics on compact (2n-2)-connected (4n-1)-manifolds. The focus will be largely topological: we will describe new constructions of these objects to which existing curvature results can be applied. The constructions are based on the technique of plumbing disc bundles. This is joint work with Diarmuid Crowley.