Abstracts (in alphabetical order):
Ilka Agricola
3-(α, δ)-Sasaki Manifolds and Strongly Positive Curvature
We consider 3-(α, δ)-Sasaki manifolds, generalizing the classic 3-Sasaki case. We show how these are
closely related to various types of quaternionic Kähler orbifolds via connections with skew-torsion.
Making use of this relation we discuss curvature operators and show that in dimension 7 many such
manifolds have strongly positive curvature. Joint work with Giulia Dileo (Bari) and Leander Stecker
(Hamburg).
Bernd Ammann
Minimal geodesics
A geodesic $c:\mathbb{R}\to M$ is called minimal if a lift to the universal covering globally minimizes
distance. On the $2$-dimensional torus with an arbitrary Riemannian metric there are uncountably many
minimal geodesics. In dimensions at least $3$, there may be very few minimal geodesics. If $M$ is
closed, Bangert has shown that the number of geometrically distinct minimal geodesics is bounded
below by the first Betti number $b_1$. In joint work with Clara Löh, we improve Bangert's lower bound
and we show that this number is at least $b_1^2+2b_1$. Our proof detects heteroclinic minimal
geodesics, which seem to have been overseen so far by classical references on this subject.
Carla Cederbaum
Black hole and photon surface uniqueness in static, asymptotically flat spacetimes: different approaches
in vacuum and electro-vacuum
It is well-known that static vacuum asymptotically flat spacetimes containing a black hole must be
isometric to the Schwarzschild spacetime. In recent years, it was shown that this result can be extended
to higher dimensions as well as to spacetimes containing so-called photon spheres and equipotential
photon surfaces. After a brief review of these facts and the underlying definitions, we will present a new
approach to proving such uniqueness results, based in part on work by Robinson. This new approach
relies on a geometric tensor serving as a suitable replacement of the Cotton tensor; this tensor was
introduced to the study of Ricci flow by Brendle. This is joint work with Albachiara Cogo, Benedito
Leandro, and João Paula dos Santos
Anand Dessai
Even dimensional manifolds which carry infinitely many geometrically distinct metrics of positive Ricci
curvature
Let us call two metrics of positive Ricci curvature on a closed manifold M geometrically distinct if they
represent different components in the moduli space of all Ric>0-metrics of M. Manifolds which carry
infinitely many geometrically distinct metrics of positive Ricci curvature have been exhibited in odd
dimensions. In my talk I will first briefly survey the known results. Then I will explain how in favorable
situations these results can be "lifted" to obtain even dimensional manifolds which carry infinitely many
geometrically distinct metrics of positive Ricci curvature. Eta invariants are used to distinguish these
metrics.
Xin Fu
Kähler-Einstein metric near an isolated log canonical singularity
Log canonical singularity is one of the most important class of singularity in higher dimensional birational
geometry. It also plays an important role in the compactification of canonical polarized variety (both
algebraic and differential geometrically.) In this talk, we will discuss the geometry of Kahler-Einstein
metric with negative scalar curvature near an isolated log canonical surface singularity. The talk is based
on joint works with Datar, Hein, Jiang and Song.
Bernhard Hanke
Scalar curvature rigidity of warped product metrics
We show scalar-mean curvature rigidity of warped products of round spheres of dimension at least $2$
over compact intervals equipped with strictly log-concave warping functions. This generalizes earlier
results of Cecchini-Zeidler to all dimensions.
Moreover, we show scalar curvature rigidity of round spheres of dimension at least $3$ minus two
antipodal points, thus resolving a problem in Gromov's ``Four Lectures'' in all dimensions.
Our arguments are based on spinor geometry.
(Joint work with Christian Bär, Simon Brendle and Yipeng Wang.)
Ramiro Lafuente
Ancient Ricci flows of bounded girth
We will explain how to construct, for each n\geq 3, a new "pancake-like" ancient Ricci flow on S^n with
positive curvature operator. The solution has cohomogeneity-one symmetry, its construction is based on
PDE methods, and it is inspired by a Mean Curvature Flow analogue. We will also discuss detailed
asymptotics and some open questions. This is based on joint work with Theodora Bourni, Tim Buttsworth
and Mat Langford.
Yi Lai
O(2)-symmetry of 3D steady gradient Ricci solitons
For any 3D steady gradient Ricci soliton with positive curvature, if it is asymptotic to a ray we prove that it
must be isometric to the Bryant soliton. Otherwise, it is asymptotic to a sector and called a flying wing.
We show that all flying wings are O(2)-symmetric. Hence, all 3D steady gradient Ricci solitons are
O(2)-symmetric.
Claude LeBrun
Einstein 4-Manifolds, Weyl Curvature, and Orbifold Limits
If we are given a sequence (Mj , gj) of smooth compact connected Einstein 4-manifolds of fixed Einstein
constant λ > 0 and fixed Euler characteristic whose volumes are uniformly bounded away from zero, the
celebrated work of Anderson and Bando-Kasue-Nakjima shows that there is a subsequence (Mji , gji )
that converges, in the Gromov-Hausdorff sense, to a 4-dimensional Einstein orbifold (X, h) with the same
Einstein constant λ. However, not every 4-dimensional positive Einstein orbifold (X, h) arises as such a
limit. Indeed, if (X, h) is a K ̈ ahler-Einstein orbifold with λ > 0, then, provided some reasonable auxiliary
hypotheses are satisfied, we show that this happens if and only if (X, h) is the limit of a sequence of
smooth K ̈ ahler-Einstein metrics. In particular, it then follows that (X, h) must be one of the orbifold limits
enumerated by Odaka-Spotti-Sun. The results described in this talk are joint with Tristan Ozuch. My aim
will be to first carefully explain the auxiliary assumptions we need to arrive at the desired conclusion, and
then explain a few of the key ideas used in the proof.
John Lott
Scalar curvature on noncompact manifolds
We discuss topological obstructions for a noncompact manifold to have a complete Riemannian metric
with (nonuniformly) positive scalar curvature. There are possible applications to simplicial volume of
closed manifolds.
Ilaria Mondello
Limits of manifolds with a Kato bound on the Ricci curvature
In this talk I will present some recent results obtained in collaboration with G. Carron and D. Tewodrose
about the structure of Gromov-Hausdorff limits of manifolds with Ricci curvature satisfying a Kato integral
bound. Under this weak condition, we prove a regularity theory in the spirit of Cheeger and Colding's
work on Ricci limits.
Lawrence Mouillé
Positive intermediate Ricci curvature with maximal symmetry rank
In foundational work for studying positively curved spaces with symmetries, Grove and Searle
established a diffeomorphism classification of n-manifolds with positive sectional curvature that have
maximal symmetry rank (i.e. isometry group with rank equal to that of O(n+1)). In this talk, I will present
work on generalizing their rigidity result to manifolds with positive intermediate Ricci curvature. The key
tools that we develop are generalizations of the Berger-Sugahara isotropy rank lemma and Wilking's
connectedness lemma for fixed point sets to this weaker curvature condition. This talk is based on joint
work with Lee Kennard.
Aaron Naber
Fundamental Groups and the Milnor Conjecture
Milnor conjectured that if (M,g) is a Riemannian manifold with Rc>=0, then its fundamental group is
finitely generated. Together with Brue and Semola we provide a counterexample. Specifically, we
construct a manifold M^7 with Rc>=0 whose fundamental group is Q/Z. This involves a new topological
construction of spaces with infinitely generated fundamental groups, and a careful analysis of the
relationship between Ricci curvature and mapping class groups.
Lei Ni
Locally Chern homogeneous spaces
Ambrose-Singer gave a characterization on when a complete Riemannian manifold is locally
homogeneous via the existence of the so-called Ambrose-Singer connection. In this talk I shall explain a
classification result proved jointly with F. Zheng on Hermitian manifolds when the Chern connection is
Ambrose-Singer. The universal cover of such a manifold is the product of a complex Lie group and
Hermitian symmetric spaces. The proof exploits the existence of certain `symplectic' holomorphic
2-forms and the holonomy systems introduced by J. Simons.
Tristan Ozuch
4-dimensional specific aspects of Ricci flows
Ricci flow has been extensively studied, and most results are either specific to dimension 3 or valid in
any dimension. However, given the potential topological applications, a theory specific to the
4-dimensional situation is desirable. In this discussion, I will present tools and techniques that are unique
to dimension 4. Together with A. Deruelle, we introduce a notion of stability for orbifold singularities. This notion helps to explain the formation of orbifold singularities along Ricci flow and lets us construct large classes of
ancient Ricci flows. In collaboration with K. Naff, we utilize self-duality in dimension 4 to simplify the
evolution equations of curvature. Among other things, this approach leads us to uncover intriguing links
between Ricci flow and Yang-Mills flow.
Anton Petrunin
Gromov's tori are optimal
We give an optimal bound on normal curvatures of immersed n-torus in a Euclidean ball of large
dimension.
Philipp Reiser
Positive intermediate Ricci curvature, surgery and Gromov's Betti number bound
Positive intermediate Ricci curvature is a family of interpolating curvature conditions between positive
sectional and positive Ricci curvature. While many results that hold for positive sectional or positive Ricci
curvature have been extended to these intermediate conditions, only relatively few examples are known
so far. In this talk I will present several extensions of construction techniques from positive Ricci
curvature to these curvature conditions, such as surgery and bundle techniques. As an application we
obtain a large class of new examples of manifolds with a metric of positive intermediate Ricci curvature,
including all homotopy spheres that bound a parallelizable manifold, and show that Gromov's Betti
number bound for manifolds of non-negative sectional curvature does not hold from positive Ricci
curvature up to roughly halfway towards positive sectional curvature. This is joint work with David Wraith.
Regina Rotman
Length of a shortest closed geodesic on a closed 3-manifold with positive scalar curvature
Let M be a closed Riemannian 3-manifold with scalar curvature bounded below by some positive
constant k. We will prove that there exists a closed non-trivial geodesic on M of length at most
$\frac{c}{\sqrt{k}}$. (Joint with Y. Liokumovich, D. Maximo.)
Daniele Semola
The metric measure boundary of non smooth spaces with lower Ricci bounds
The metric measure boundary of a metric measure space measures the average distortion of the
volumes of small balls with respect to the corresponding Euclidean values, at the first order. A few years
ago, A. Lytchak, V. Kapovitch and A. Petrunin found an interesting connection between the vanishing of
the metric measure boundary and the geodesic flow on Alexandrov spaces and conjectured that the
metric measure boundary of any Alexandrov space with empty boundary should vanish. I will discuss
joint work with E. Bruè and A. Mondino where we established the conjecture in the more general case of
metric measure spaces with lower Ricci Curvature bounds.
Artemis A. Vogiatzi
Singularity models for high codimension mean curvature flow in Riemannian manifolds
In this talk, we show that under a quadratic curvature pinching hypothesis, in regions of high curvature,
the submanifold under the mean curvature flow becomes approximately codimension one in a
quantifiable way. This enables us to prove that at a singularity, there exists a rescaling that converges to
a smooth codimension-one limiting flow in Euclidean space, which is weakly convex and either moves by
translation or is a self-shrinker.
Matthias Wink
Einstein metrics on the Ten-Sphere
In this talk we give an introduction to the topic of Einstein metrics on spheres. In particular, we prove the existence of three non-round Einstein metrics with positive scalar curvature on $S^{10}.$ Previously, the only even-dimensional spheres known to admit non-round Einstein metrics were $S^6$ and $S^8.$ This talk is based on joint work with Jan Nienhaus.
Sergio Zamorra Barrera
Lower Ricci bounds and fundamental group. Recent work on smooth and singular spaces
On Riemannian manifolds, it is understood how lower Ricci curvature bounds influence the fundamental group. Recently Jikang Wang proved that RCD spaces are semi-locally-simply-connected, allowing us to extend this understanding to the non-smooth realm. We will also present recent work on the behavior of the first Betti number under collapse, new even in the smooth setting. Joint work with Qin Deng, Jaime Santos, and Xinrui Zhao.
