Abstracts

Manuel Amann, Karlsruher Institut für Technologie
Topological properties of positively curved manifolds with symmetry
Manifolds admitting metrics of positive sectional curvature are conjectured to have a very rigid homotopical structure and, in particular, comparatively small Euler characteristics. However, this structure is still highly speculative and best results in this direction are known under the assumption of large isometric torus actions.

In this talk I shall present joint work with Lee Kennard on new upper bounds on the Euler characteristic of a closed manifold which admits a metric of positive curvature and an isometric torus action. The results allow one to find obstructions to symmetric spaces, manifold products and connected sums admitting positively curved metrics with symmetry, thus providing evidence for a conjecture of Hopf.

I shall discuss how the additional assumption of rational ellipticity comes in naturally to yield further results like, for example, a confirmation of the Halperin conjecture in this setting. Furthermore, vanishing properties of the elliptic genus of positively curved manifolds with symmetry will be derived.


Richard Bamler, Stanford University
There are finitely many surgeries in Perelman's Ricci flow
Although the Ricci flow with surgery has been used by Perelman to solve the Poincaré and Geometrization Conjectures, some of its basic properties are still unknown. For example it has been an open question whether the surgeries eventually stop to occur (i.e. whether there are finitely many surgeries) and whether the full geometric decomposition of the underlying manifold is exhibited by the flow as $t \to \infty$.

In this talk I will show that the number of surgeries is indeed finite and that the curvature is globally bounded by $C t^{-1}$ for large $t$. Using this curvature bound it is possible to give a more precise picture of the long-time behavior of the flow.

The proof of this result builds on a previous theorem in which I established finiteness of the surgeries as well as the curvature bound under a certain topological assumption. I will give a very brief overview of the proof of this theorem and explain why this extra assumption was crucial. Next, I will present new tools which enabled me to remove this assumption completely.


Nicola Gigli, Université de Nice
Analytic tools for the study of spaces with Ricci curvature bounded from below.
I will show that every metric measure space admits a weak first order differentiable structure which allows to integrate by parts. I will present some use of this structure in relation with the Lott-Sturm-Villani synthetic notion of lower bound on the Ricci curvature. In particular, I will discuss the Cheeger-Colding-Gromoll splitting theorem in such context.


John Harvey, University of Notre Dame
Positively curved Alexandrov spaces of maximal symmetry rank
This talk is based on joint work with Catherine Searle. We develop two new tools for use in Alexandrov geometry: a theory of ramified orientable double covers and a particularly useful version of the Slice Theorem for actions of compact Lie groups. These tools are applied to the classification of compact positively curved Alexandrov spaces with maximal symmetry rank.


Robert Haslhofer, Courant Institute, NYU
Mean curvature flow of mean convex hypersurfaces
In the last 15 years, White and Huisken-Sinestrari developed a far-reaching structure theory for the mean curvature flow of mean convex hypersurfaces. Their papers provide a package of estimates and structural results that yield a precise description of singularities and of high curvature regions in a mean convex flow. In this talk, I will describe a new treatment of the theory of mean convex (and k-convex) flows. This includes: (1) an estimate for derivatives of curvatures, (2) a convexity estimate, (3) a cylindrical estimate, (4) a global convergence theorem, (5) a structure theorem for ancient solutions, and (6) a partial regularity theorem. Our new proofs are both more elementary and substantially shorter than the original arguments. Our estimates are local and universal. A key ingredient in our new approach is the new non-collapsing result of Andrews. Some parts are also inspired by the work of Perelman. This is joint work with Bruce Kleiner.


Hans-Joachim Hein, Imperial College London
Asymptotically cylindrical Calabi-Yau manifolds
I will talk about complete noncompact Ricci-flat Kähler manifolds (M, g) that are asymptotic to cylinders at infinity. The main result roughly states that M can be written as X\D, where X is a compact projective algebraic variety and D is an anticanonical hypersurface in X with trivial normal bundle. Conversely every such pair (X, D) gives rise to a Ricci-flat space (M, g). Joint work with Mark Haskins and Johannes Nordström.


Sebastian Hoelzel, Westfälische Wilhelms-Universität, Münster
Surgery stable curvature conditions
We give a simple criterion for a pointwise curvature condition to be stable under surgeries of certain codimensions. This is used to show the following: Any simply-connected manifold of dimension at least 5 which is either spin with vanishing alpha-invariant or else is non-spin admits metrics with pointwise almost nonnegative curvature operator.


Brett Kotschwar, Arizona State University
Uniqueness of asymptotically conical shrinking Ricci solitons
Shrinking Ricci solitons are generalizations of positive Einstein manifolds, and occupy a central place in the analysis of singularities of the Ricci flow. At present, all known nontrivial (i.e., non-product) examples of complete noncompact gradient shrinking solitons have a cone structure at infinity. I will describe recent joint work with Lu Wang in which we prove that such a structure is rigid in the sense that if two shrinking gradient Ricci solitons are asymptotic to the same cone at infinity along some end of each, then the solitons must actually be isometric on some neighborhoods of infinity on the ends. We further show that a shrinking soliton that is asymptotic to a rotationally symmetric cone along some end must actually be rotationally symmetric, and hence complete if and only if it is the flat Gaussian soliton on R^n.


John Lott, University of California, Berkeley
Collapsing with a lower bound on the curvature operator
Cheeger and Gromov showed that F-structures are related to collapse with a double-sided curvature bound. We define fibered F-structures and extend some of the Cheeger-Gromov results to the setting of collapse with a lower bound on the curvature operator.



Alexander Lytchak, Universität zu Köln
Riemannian foliations on spaces with very simple or no topology
In the talk I will explain the following related results. Any Riemannian foliation on a contractible manifold has closed leaves. Moreover, it is "often" given by a global Riemannian submersion. Any Riemannian foliation of a sphere with leaves of dimension at least 7 is a simple foliation. This result provides a final piece in the classification of Riemannian foliations on round spheres, due to Ghys, Gromoll, Grove and Wilking.


Anton Petrunin, Pennsylvania State University
Number of collisions of n balls on the infinite billiard table after Burago--Ferleger--Kononenko
Consider a system of n hard balls moving freely and colliding elastically in Euclidean space. By the theorem of D. Burago, S Ferleger and A. Kononenko, there is an upper bound for the number of collisions in such a system. We present a simplified proof of this theorem.


Conrad Plaut, University of Tennessee
Some Applications of Discrete Homotopy Theory
Discrete homotopies were introduced by Berestovskii and Plaut for topological groups and uniform space, and pursued further for metric (especially geodesic) spaces by Plaut and Jay Wilkins, who is a co-author for this talk. Continuous paths and homotopies are replaced by discrete chains and homotopies that detect "holes" at a given scale, producing corresponding covering spaces and deck groups. We will discuss the background and basic ideas, including homotopy critical values, essential circles and triads. Our first application is a "curvature-free" generalization and extension of a theorem of Gromov on the number of generators of the fundamental group of a compact Riemannian manifold. As a corollary we obtain a fundamental group finiteness theorem for geodesic spaces (new even for Riemannian manifolds) that replaces the curvature and volume conditions of Anderson and the 1-systole bound of Shen-Wei, by more general geometric hypotheses implied by these conditions. As a second application we define a new class of covering maps called circle covers that extends the notion of delta-cover of Sormani-Wei, that has the advantage of being "closed" with respect to Gromov-Hausdorff Convergence. This allows us to completely understand limits of delta-covers. There are various interesting open questions, including the degree to which the concepts, defined by metrics, are in fact dependent only on the underlying topology.


Viktor Schroeder, Universität Zürich
Moebius characterizations of the boundaries of complex hyperbolic spaces
There is a deep and well known relationship between the geometry of the classical hyperbolic space and the Moebius geometry of its boundary at infinity. We generalize this relation to more general negatively curved spaces. In particular we are interested in a characterization of the boundaries of rank one symmetric spaces in terms of Moebius geometry. This is joined work with Thomas Foertsch and Sergei Buyalo.


Catherine Searle, Oregon State University
Lifting positive Ricci curvature
We show how to lift positive Ricci and almost nonnegative curvatures from an orbit space $M/G$ to the corresponding $G$--manifold, $M$. We apply the results to get new examples of Riemannian manfiolds that satisfy both curvature conditions simultaneously. This is joint work with Fred Wilhelm.


Wolfgang Spindeler, Westfälische Wilhelms-Universität, Münster
A decomposition theorem for nonnegatively curved fixed point homogenous manifolds
Let a compact Lie Group $G$ act by isometries on a Riemannian manifold $(M,g)$ with quotient space $M/G$. $M$ is called fixed point homogenous if $G$ acts transitively on a normal sphere to some fixed point component $F$. Equivalently, considering $F$ as a subset of $M/G$, the codimension of $F$ in $M/G$ is one. K. Grove and C. Searle gave an equivariant classification of closed fixed point homogenous Riemannian manifolds of positive curvature. One important step to obtain this classification is to show that there is a unique orbit $G \ast p \subset M$ of maximal distance to $F$ and $M$ is diffeomorphic to the unit normal bundles $D(F)$ and $D(G \ast p)$ of $F$ and $G \ast p$ glued together along their boundaries.

Following this ideas we show that for a closed nonnegatively curved fixed point homogenous manifold $(M,g)$ with fixed point component $F$ of maximal dimension there exists a $G$-invariant smooth submanifold $N \subset M$ such that $M$ is diffeomorphic to the unit normal bundles $D(F)$ and $D(N)$ of $F$ and $N$ glued together along their boundaries. This result is obtained via the geometric properties of the space $M/G$ which is an Alexandrov space of lower curvature bound $0$ with nonempty boundary.


Valentino Tosatti, Northwestern University
Collapsing of Ricci-flat Calabi-Yau manifolds
We will discuss the problem of understanding how Ricci-flat Calabi-Yau manifolds collapse to lower-dimensional spaces, and present some results in this direction which are joint work with Mark Gross and Yuguang Zhang.


Mark Walsh, Wichita State University
Loop Spaces, operads and positive scalar curvature
Recently, there have been a number of interesting developments concerning the problem of understanding the space of metrics of positive scalar curvature on a smooth manifold. In this talk, I will discuss a new result which shows that in the case when the underlying manifold is a sphere of dimension $n$, where $n$ is at least three but not equal to four, the space of metrics of positive scalar curvature is weakly homotopy equivalent to an $n$-fold loop space. This result makes considerable use of the ``recognition principle" of Boardman-Vogt and May as well as a recent theorem of Botvinnik on concordance and isotopy in the space of metrics of positive scalar curvature.


Michael Wiemeler, Karlsruher Institut für Technologie
Non-negatively curved torus manifolds
The classification of Riemannian manifolds with lower curvature bounds has a long history in geometry. We discuss this classification in the context of torus manifolds, i.e. even dimensional manifolds $M$ which admit effective actions of tori $T$ such that $2\dim T=\dim M$ and $M^T \neq\emptyset$. It has been shown by Grove and Searle that a simply connected torus manifold which admits an invariant metric of positive sectional curvature is diffeomorphic to a sphere or a complex projective space. We show that a simply connected torus manifold with $H^{\text{odd}}(M;\mathbb{Z})=0$ which admits an invariant metric of non-negative curvature is homeomorphic to a quotient of a free linear torus action on a product of spheres. All these quotients admit invariant metrics of non-negative curvature.


Wolfgang Ziller, University of Pennsylvania
Positively curved submanifolds of euclidean space
A. Weinstein and D.Moore showed that a submanifold of euclidean space, immersed in codimension 2, whose induced metric has positive sectional curvature, must be diffeomorphic to a sphere. We discuss the case of non-negative curvature in codimension 2 and make some comments about the case codimension 3. The techniques are a combination of Ricci flow and submanifold geometry. This is joint work with Luis Florit.