##### Schedule

##### Abstracts

## Stephanie Alexander, University of Illinois Urbana-Champaign

Convex Functions and Comparison Geometry of Spacetimes

We study classically convex and "spacetime convex" functions (introduced by Gibbons-Ishibashi, 2001) in space-time geometry. Using classically convex functions, we give a geometric-topological approach to geodesic connectedness of space-times. As a corollary, timelike strictly convex hypersurfaces of Minkowski space are geodesically connected. Turning to spacetime convex functions, we show they are fundamentally related to curvature bounds of the form $R \leq K$ (space-like sectional curvatures $\leq K$, timelike sectional curvatures $\geq K$). These bounds were introduced and applied by Andersson-Howard, 1998, and characterized by signed triangle comparisons by Alexander-Bishop, 2008. The connection with curvature bounds allows us to identify many space-times that support spacetime convex functions. Such functions rule out, for example, closed marginally inner and outer trapped surfaces. This is joint work with William Karr.

## Fuquan Fang, Capital Normal University, Beijing

Reflections, polar actions in non-negative curvature

In this talk, I will start from an introduction to reflection groups on Riemannian manifolds with non-negative curvature, certain rigidity theorems will be explained in this context. Based on these structures, we consider polar actions on manifolds with non-negative curvature, several theorems on the structures of polar manifolds will be introduced. This is based on joint works with Karsten Grove.

## Ursula Hamenstädt, University of Bonn

A variation of Gromov-Thurston rigidity

The rigidity theorem of Gromov an Thurston states that the degree of a map $f\colon M\to N$ between hyperbolic manifolds is not bigger than the volume ratio of M and N, with equality if and only if $f$ is homotopic to an
isometric covering. We discuss a version of this result where N is allowed to be a $K(\pi,1)$ of a hyperbolic group and volume is replaced by simplicial volume.

## Nancy Hingston, The College of New Jersey

String Topology, Index Growth, and Closed Geodesics

Morse theory provides a bridge between the geometry of closed geodesics on a compact Riemannian manifold $M$, and the topology of the free loop space of $M$. Chas and Sullivan introduced the algebraic structure on loop space homology known as string topology. I will describe string topology from the point of view of geometry and discuss its relationship to closed geodesics and index growth. Mark Goresky, Alexandru Oancea, Hans-Bert Rademacher, and Nathalie Wahl are collaborators.

## Martin Kerin, WWU Münster

Non-negative curvature on exotic spheres

Since their discovery, there has been much interest in the question of precisely which exotic spheres admit a metric with non-negative sectional curvature. In dimension $7$, Gromoll and Meyer found the first such example. It was subsequently shown by Grove and Ziller that all of the Milnor spheres admit non-negative curvature. In this talk, it will be demonstrated that the remaining exotic $7$-spheres also admit non-negative curvature. This is joint work with K. Shankar and S. Goette.

## Nan Li, CUNY - CityTech

On the tangent cones in collapsed limit spaces with lower Ricci curvature bound

We show that if $X$ is a limit of $n$-dimensional Riemannian manifolds with Ricci curvature bounded below and $\gamma$ is a limit geodesic in $X$, then along the interior of $\gamma$, same scale measure metric tangent cones are Holder continuous with respect to measured Gromov–Hausdorff topology and have the same dimension in the sense of Colding-Naber. This is a joint work with Vitali Kapovitch.

## Ayato Mitsuishi, Gakushuin University

Local Lipschitz contractibility of Alexandrov spaces and its applications

In the collapsing theory, Perelman's stability theorem is very important: it states that if two compact Alexandrov spaces of the same dimension are close to each other in the Gromov-Hausdorff topology, then they are homeomorphic. Perelman further stated that the statement is also true when "homeomorphic" is replaced by "bi-Lipschitz homeomorphic". However, this stronger claim was not proven. In the present talk, I will announce a partial answer of the claim, obtained by me and Yamaguchi (Kyoto).

## Ilaria Mondello, Institut de Mathématiques de Jussieu

Geometry and analysis on stratified spaces

Stratified spaces are singular metric spaces arising naturally in differential geometry as quotients or limits of smooth manifolds; they have also been studied from the topological and analytical points of view. This talk is devoted to show how to obtain in the singular setting some theorems which, restricted to the smooth case, recover classical results of Riemannian geometry and geometric analysis (Obata-Lichnerowicz and Myers diameter theorem, Sobolev inequalities...). Such theorems can also be applied in order to study the existence of a conformal metric of constant scalar curvature on a stratified space.

## Xuan Hien Nguyen, Iowa State University

Mean curvature flow of entire graphs evolving away from the heat flow

We present two initial graphs over the entire $\mathbb R^n$, $n \geq 2$ for which the mean curvature flow behaves differently from the heat flow. In the first example, the two flows stabilize at different heights. With our second example, the mean curvature flow oscillates indefinitely while the heat flow stabilizes. These results highlight the difference between dimensions $n \geq 2$ and dimension $n=1$, where Nara and Taniguchi proved that bounded smooth entire graphs over the line converge to solutions to the heat equation with the same initial data.

## Raquel Perales, MSRI/UNAM

Limits of Manifolds with Ricci Curvature and Mean Curvature Bounds

We consider smooth Riemannian manifolds with nonnegative Ricci curvature
and smooth boundary. First we prove a global Laplace comparison theorem
in the barrier sense for the distance to the boundary. We apply this theorem
to obtain volume estimates of the manifold and of regions of the manifold near
the boundary depending upon an upper bound on the area and on the mean curvature of the boundary. We prove that families of oriented manifolds with uniform bounds of this type are compact with respect to the Intrinsic Flat distance.

## Marco Radeschi, WWU Münster

Invariant Theory of singular Riemannian foliations

Singular Riemannian foliations are metric structures on Riemannian manifolds which generalize isometric group actions. In this talk, I will present some recent results and work in progress, directed toward understanding the algebra of functions which are constant along the leaves of a singular Riemannian foliation. In the particular case of foliations in Euclidean space, the results generalize well-known theorems in the classical Invariant theory of representations of reductive groups.

## Priyanka Rajan, University of California, Riverside

Fake 13-Projective Spaces

In this talk, I am going to outline the proof that some embedded standard $13$-spheres in Shimada’s exotic $15$-spheres have $\mathbb Z_2$ quotient spaces, $P^{13}$s, that are fake real $13$-dimensional projective spaces, i.e., they are homotopy equivalent, but not diffeomorphic to the standard $\mathbb R P^{13}$.

## Anna Siffert, Max Planck Institute for Mathematics

Harmonic maps of spheres and special orthogonal groups

We develop the theory of equivariant harmonic self-maps of compact construct new harmonic self-maps of spheres and special orthogonal groups with nontrivial degree. Parts of this are joint work with Thomas Püttmann.

## Christina Sormani, City University of New York

Intrinsic Flat Convergence of manifolds with nonnegative scalar curvature

While Gromov proved sequences of manifolds with nonnegative Ricci curvature have subsequences converging with respect to his intrinsic Hausdorff convergence, Ilmanen's famous example demonstrates no such theorem holds for sequences of manifolds with nonnegative scalar curvature. In joint work with Wenger, the intrinsic flat distance was introduced applying techniques of Ambrosio-Kirchheim. In joint work with Lee, with LeFloch, with Stavrov and with Huang-Lee this new distance has been applied to study sequences of manifolds with nonnegative scalar curvature applying work of Wenger, Portegies and Lakzian. A recent survey with open questions is available

here.

## Chuu-Lian Terng, University of California, Irvine

What is a Canonical Form? Examples and Applications

In this talk, we (i) review Palais' notion of an H-slice
(i.e., a canonical form) for a group action, (ii) give some examples
-- polar actions, the action of the diffeomorphism group on the space
of Riemannian metrics, moving frames for curves in homogeneous spaces,
and (iii) explain some applications in differential geometry and the
theory of soliton equations.

## Wouter Van Limbeek, University of Michigan

Symmetry gaps in Riemannian geometry and minimal orbifolds

In 1893 Hurwitz showed that a hyperbolic surface of genus at least $2$ has isometry group of order at most $84(g-1)$. Do such bounds on the order of isometry groups exist more generally? It was conjectured by Farb-Weinberger that this is the case for certain aspherical manifolds. In this spirit we prove that the size of the isometry group of an arbitrary closed manifold is bounded in terms of certain geometric quantities (such as curvature and volume), unless the manifold admits an action by a compact connected Lie group. We give two applications of this result: First we characterize locally symmetric spaces among all Riemannian manifolds, and secondly, we generalize results of Kazhdan-Margulis and Gromov on the existence of minimal quotients of locally symmetric spaces and negatively curved manifolds.

## Burkhard Wilking, WWU Münster

Computing the Euler characteristic of positively curved manifolds under logarithmic symmetry assumptions

We show that the Euler characteristic of a positively curved n manifold $M$ coincides with the Euler characteristic of an $n$-dimensional compact rank $1$ symmetric space provided that the rank of the isometry group of $M$ is larger than $3\log_2 n$.

## Wolfgang Ziller, University of Pennsylvania

On the Nomizu conjecture and Riemannian graph manifolds

In many geometric problems the curvature tensor of a Riemannian manifold has large nullity space. We show that, under certain regularity assumptions, a Riemannian manifold with almost maximal nullity is isometric to an $n$-dimensional graph manifold. As a consequence we show that Nomizu's conjecture holds for finite volume manifolds. This is joint work with Luis Florit.