Abstracts

Abstracts of all lectures are needed for our records. Several participants have also expressed interest in having them available by the beginning of the meeting. Please email us your abstract as soon as possible, not to arrive later than Monday, July 18.

Christian Bär, Universität Potsdam
Asymptotic heat kernel expansion in the semi-classical limit
Let $H_h = h^2 L +V$ where $L$ is a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold and $V$ is a symmetric endomorphism field. We derive an asymptotic expansion for the heat kernel of $H_h$ as $h \to 0$. As a consequence we get an asymptotic expansion for the quantum partition function and we see that it is asymptotic to the classical partition function. Moreover, we show how to bound the quantum partition function for positive $h$ by the classical partition function.


Stefan Bechtluft-Sachs, National University of Ireland, Maynooth
Positive Ricci Curvature on G-Manifolds with Finitely Many Non-Principal Orbits
Positive Ricci curvature on G-manifolds of cohomogeneity 0 or 1 is controlled by the fundamental group: Such a manifold admits an invariant metric of positive Ricci curvature if and only if its fundamental group is finite. It is also known that such a result can not hold in cohomogeneity 4 or more. G-manifolds with finitely many non-principal orbits directly generalize the cohomogeneity 1 case (where there are precisely 2 or no non-principal orbits). We describe the structure of such manifolds and then give a construction of positive Ricci curvature metrics. This is joint work with David Wraith.


Esther Cabezas-Rivas, Westfälische Wilhelms-Universität Münster
How to produce a Ricci Flow via Cheeger-Gromoll exhaustion
We prove short time existence for the Ricci flow on open manifolds without requiring upper bounds on the curvature. We ask the starting manifold M to have non-negative complex sectional curvature to ensure that M admits a Cheeger-Gromoll convex exhaustion, which is the main tool to prove our theorem. Furthermore, we find a optimal volume growth condition which gives long time existence, and we construct an explicit example of an immortal non-negatively curved solution of the Ricci flow with unbounded curvature for all time.


Fernando Galaz-Garcia, Westfälische Wilhelms-Universität Münster
Cohomogeneity-two torus actions on nonnegatively curved manifolds of low dimension
Let M be a simply-connected Riemannian n-manifold with nonnegative sectional curvature. If n is at most 9, it is known that a torus acting effectively and isometrically on M must be of dimension at most 2n/3. In this talk I will discuss the topological and equivariant classification of simply-connected 4- and 5-manifolds with nonnegative sectional curvature supporting the largest possible torus actions corresponding, in dimension 4, to an action of a 2-torus, and in dimension 5, to an action of a 3-torus.


Robert Haslhofer, ETH Zürich
Stability of Ricci-flat spaces
I will give an overview about different aspects of the stability of Ricci-flat spaces. In the case of compact manifolds, I will discuss a Lojasiewicz-Simon inequality and its dynamical implications for the Ricci flow. In the noncompact case, I will introduce a renormalized Perelman-functional, discuss the stability inequality for Ricci-flat cones, and explain an intriguing relationship with the ADM-mass from general relativity.


Chenxu He, Lehigh University
Warped product Einstein structures
We consider the question about when a fixed Riemannian manifold is the base of a warped product Einstein metric. This problem has been completely solved when the base is 1 or 2-dimensional and much progress has been made in higher dimensions as well. There are also many interesting extensions to the case where the base might have boundary and when we allow for warping functions that change sign. When there are more than one warping function for a fixed base manifold, they induce a natural stratification on the manifold. Using this stratification we are able to show some rigidity results. This is joint work with Peter Petersen at UCLA and William Wylie at UPenn.


Michael Jablonski, University of Oklahoma
Ricci soliton solvmanifolds are algebraic
In this talk we investigate the property that a Riemannian metric is both homogeneous and a Ricci soliton. In the presence of a transitive solvable group of isometries, we show that such a metric is isometric to a solvsoliton; i.e. its Ricci tensor is of the form Ric = c Id + D, where c is a real number and D is a derivation of a solvable Lie algebra.


Lee Kennard, University of Pennsylvania
Periodic cohomology in positive curvature
Hopf and Chern made conjectures concerning the Euler characteristics and fundamental groups, respectively, of manifolds admitting positively curved metrics. We discuss progress on these conjectures in the presence of symmetry. The main new idea is to study the action of the Steenrod algebra on the periodic cohomology rings that arise in the setting of Wilking's connectedness theorem.


John Lott, University of California, Berkeley
Geometrization of orbifolds via Ricci flow
A three-dimensional compact orbifold (with no bad suborbifolds) is known to have a geometric decomposition from the work of Perelman along with earlier work of Boileau-Leeb-Porti/Cooper-Hodgson-Kerckhoff. I'll describe a unified proof of the geometrization of orbifolds, using Ricci flow. The emphasis will be on the aspects that are particular to orbifolds. This is joint work with Bruce Kleiner.


Alexander Lytchak, Westfälische Wilhelms-Universität Münster
Isometries between quotients
Abstract


Anton Petrunin, Pennsylvania State University
Sweeping curvature out and more.
This is a work in progress with Dmitri Panov. We observe that the maximal open set of constant curvature k in a Riemannian manifold with curvature bounded below or above by k has a convexity type property, which we call "two-convexity". This statement is used to prove a number of rigidity statements in comparison geometry. We also discuss a related problem for polyhedral spaces which is linked to a problem about complex hyperplane arrangements.


Thomas Püttmann, Ruhr-Universität Bochum
On the commutator of unit quaternions and exotic actions
The $k$-th power of the commutator of unit quaternions is null-homotopic if and only if $k$ is divisible by $12$. The main subject of this talk is the construction of concrete null-homotopies of the $12j$-th powers of the commutator of unit quaternions. Such null-homotopies yield trivializations of the $12j$-th principal $\mathbb{S}^3$-bundles over $\mathbb{S}^7$. In combination with the Gromoll-Meyer action of $\mathbb{S}^3$ on $\mathrm{Sp}(2)$ and its generalization to $\mathbb{S}^3$-actions on the principal $\mathbb{S}^3$-bundles over $\mathbb{S}^7$ by C.~Duran, A.~Rigas and the author, we obtain free non-isometric $\mathbb{S}^3$-actions on $\mathbb{S}^7\times\mathbb{S}^3$ whose quotients are exotic $7$-spheres $\Sigma^7_{12j}$. These exotic spheres form a subgroup of order $7$ in the group of orientation preserving diffeomorphism classes of homotopy $7$-spheres $\Theta_7 \approx \mathbb{Z}_{28}$.


Catherine Searle, UNAM Cuernavaca
Non-negatively curved 5-manifolds with almost maximal symmetry rank
We show that a closed, simply-connected, non-negatively curved $5$-manifold admitting an effective, isometric $T^2$ action is diffeomorphic to one of $S^5, S^3\times S^2$, $S^3\tilde{\times} S^2$ or the Wu manifold $SU(3)/SO(3)$. This is joint work with Fernando Galaz-Garcia.


Gudlaugur Thorbergsson, Universität Köln
Positively curved polar manifolds
In the talk I will report on joint work with Fuquan Fang and Karsten Grove. Our main result is that a compact simply connected positively curved polar manifold with cohomogenity at least three is equivariently diffeomorphic to a compact rank one symmetric space with an isometric polar action.


Wilderich Tuschmann, Karlsruher Institut für Technologie
Almost Nonnegative Curvature Operator
I will discuss constructions of manifolds with and obstructions to this property.