Abstracts (in alphabetical order):
Lorenzo Foscolo
Complete non-compact manifolds with holonomy G2 and geometric transitions
In this talk I will discuss the construction of families of complete non-compact Ricci-flat 7-manifolds with
maximal and non-maximal volume growth and holonomy G2 that exhibit a qualitative behaviour similar to
the one of 4-dimensional ALE and ALF metrics obtained from the Gibbons-Hawking Ansatz in
hyperkähler geometry. In the early 2000’s Atiyah-Maldacena-Vafa and Acharya, motivated by the physical duality between Type
IIA String Theory and M theory, proposed a relation between the so-called conifold transition in
Calabi-Yau geometry and the so-called G2-flop in G2 geometry. Both transitions involve distinct
asymptotically conical Ricci-flat manifolds with special holonomy desingularising the same Ricci-flat
cone. The relation between the two transitions suggested the existence of families of complete
G2-holonomy metrics with non-maximal volume growth “interpolatin” between the 7-dimensional cone
and the 6-dimensional one. Using analytic and cohomogeneity one methods and toric geometry we
verify this expectation and extend this picture to infinitely many G2 cones.
The talk is based on joint work with Haskins–Nordström and Acharya–Najjar–Svanes.
Georg Frenck
Diffeomorphisms and positive curvature
In this talk I will explain how to locate non-trivial rational homotopy groups of spaces of Riemannian metrics
satisfying lower curvature bounds. The idea is to construct fiber bundles where the fiber supports a metric
which satisfies this bound, whereas the total space does not.
Ursula Hamenstädt
Incompressible surfaces in locally symmetric manifolds
The surface subgroup question asks for the existence of a subgroup of a given group H which is
isomorphic to the fundamental group of a closed surface of genus at least 2. We begin with a survey on
recent progress on this question. We then explain the strategy to construct such surface subgroups in
any cocompact lattice of a classical simple Lie group of non-compact type different from SO(n,1) for
even n. The construction is explicit and geometric and extends earlier work of Kahn and Markovic and of
Kahn, Labourie, Mozes.
Bernhard Hanke
Boundary conditions for scalar curvature
On connected manifolds of dimension at least two and with nonempty boundary the existence of Riemannian
metrics with positive scalar curvature is unobstructed, unlike on closed manifolds. We discuss a number of
geometric boundary conditions, such as being of non-negative mean curvature, totally geodesic, doubling or
of product form near the boundary, which lead to nontrivial existence and classification results for positive
scalar curvature metrics. To this end, we formulate a general deformation principle that implies that the
relaxation of boundary conditions often leads to weak homotopy equivalences of spaces of positive scalar
curvature metrics. However, this is not true for the condition of being of product form near the boundary,
which is often used by topologists. Joint work with Christian Bär (Potsdam).
Lee Kennard
Torus representations, isotropy groups, and matroids
We examine torus actions on closed manifolds, and we assume that the isotropy representation at a
fixed point has the property that all isotropy groups are connected. In the presence of an invariant,
positively curved Riemannian metric, we can compute the manifold’s rational cohomology when the torus
rank is at least nine. The main work involves analyzing torus representations with connected isotropy
groups and borrowing results from the abstract, well established theory of regular matroids. This is joint
work with Michael Wiemeler and Burkhard Wilking.
Klaus Kröncke
L^p-stability and positive scalar curvature rigidity of Ricci-flat ALE manifolds
We prove stability of integrable ALE manifolds with a parallel spinor under Ricci flow, given an initial
metric which is close in $L^p\cap L^{\infty}$, for any $p\in (1,n)$, where n is the dimension of the
manifold. In particular, our result applies to all known examples of 4-dimensional gravitational instantons.
The result is obtained by a fixed point argument, based on novel estimates for the heat kernel of the
Lichnerowicz Laplacian. It allows us to give a precise description of the convergence behaviour of the
Ricci flow. Our decay rates are strong enough to prove positive scalar curvature rigidity in $L^p$, for
each $p\in [1,\frac{n}{n-2})$, generalizing a result by Appleton. This is joint work with Oliver Lindblad
Petersen.
Ramiro Lafuente
Non-compact Einstein manifolds with symmetry
This is joint work with Christoph Böhm. For Einstein manifolds with negative scalar curvature admitting a
cocompact isometric action of a connected Lie group G with smooth orbit space, we show the following
rigidity result: the nilradical N of G acts "polarly", and the N-orbits can be extended to minimal Einstein
submanifolds. As an application, we prove the Alekseevskii conjecture (1975): any homogeneous
Einstein manifold with negative scalar curvature is diffeomorphic to a Euclidean space.
Yi Lai
Steady gradient Ricci solitons with positive curvature operator
We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by
Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying
wings are collapsed. For dimension n ≥ 4, we find a family of Z2 × O(n − 1)-symmetric but
non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operator. We
show that these solitons are non-collapsed.
John Lott
Some remarks about scalar curvature
I will discuss two topics. The first is a characterization of lower scalar curvature bounds in terms of distance-decreasing maps on domains. The second is a scalar curvature measure defined on the backward limit of a Ricci flow solution.
Alexander Lytchak
CAT(0) manifolds
In the talk, I will discuss the following result obtained jointly with Koichi Nagano and Stephan Stadler:
4-dimensional CAT(0) manifolds are homeomorphic to the Euclidean space. The corresponding
statement in higher dimensions is wrong and in small dimensions well-known. The result provides a final
piece to the question of Gromov about the topology of globally non-positively curved metric spaces
homeomorphic to manifolds.
Alexander Nabutovsky
Isoperimetric inequality for Hausdorff contents and new systolic inequalities
We will discuss the isoperimetric inequality for Hausdorff contents and compact metric spaces in
(possibly infinite-dimensional) Banach spaces. We will also discuss some of its implications for systolic
geometry, in particular, systolic inequalities of a new type that are true for much wider classes of
non-simply connected Riemannian manifolds than Gromov’s classical systolic inequality. (Recall that
Gromov's systolic inequality asserts that the length of the shortest closed curve on a closed essential
Riemannian manifold M^n does not exceed c(n)vol(M^n)^{1/n}.) Joint work with Y. Liokumovich, B.
Lishak, and R. Rotman.
Lei Ni
Quadratic curvatures and compact homogeneous Kähler manifolds
Quadratic bisectional curvature has been around since 1960s. Its positivity was proposed to characterize the simply-connected homogenous Kaehler spaces by Wu-Yau-Zheng in 2009. Recently there are new notions of quadratic curvatures were proposed as perhaps the `right' candidates. I shall explain what they are, the connection with the so-called generalized Hartshorne conjecture and some progresses.
Jan Nienhaus
An improved four-periodicity theorem
Lee Kennard's four-periodicity theorem has played an important role in the recent study of closed positively curved manifolds with torus symmetry. Its main application states that whenever $N^{n-k}\to M^n$ is a submanifold with maximally connected inclusion map, then $M$ has four-periodic rational cohomology ring as long as $k\le\frac{n}{3}$. We prove the same for $k\le \frac{n}{2}$ under the additional assumption that $N$ is the fixed point set of a semi-free $S^1$ action. As this condition is automatic in most applications, as an application we reduce the necessary symmetry assumptions in some recent theorems of Kennard, Wiemeler and Wilking. The main new tool is translating the problem into a question about characteristic classes.
Jiewon Park
Canonical identification between scales on Ricci-flat manifolds
Abstract: We will study complete Ricci-flat manifolds with Euclidean volume growth. In the case when a tangent cone at infinity of the manifold has smooth cross section, the Green function for the Laplace equation can be used to define a functional which measures how fast the manifold converges to the tangent cone. Using the Łojasiewicz inequality of Colding-Minicozzi for this functional, we describe how two arbitrarily far apart scales in the manifold can be identified in a natural way. I will also discuss a matrix Harnack inequality for the Green function when there is an additional condition on sectional curvature, which is an analogue of various matrix Harnack inequalities obtained by Hamilton and Li-Cao in time-dependent settings.
Tristan Ozuch
Noncollapsed degeneration and desingularization of Einstein 4-manifolds
We study the moduli space of unit-volume Einstein 4-manifolds near its finite-distance boundary, that is,
the noncollapsed singularity formation. We prove that any smooth Einstein 4-manifold close to a singular
one in a mere Gromov-Hausdorff (GH) sense is the result of a gluing-perturbation procedure that we
develop and which handles the presence of multiple trees of singularities at arbitrary scales. This sheds
some light on the structure of the moduli space and lets us show that spherical and hyperbolic orbifolds
which are Einstein in a synthetic sense cannot be GH-approximated by smooth Einstein metrics.
Regina Rotman
The length of the shortest closed geodesic on closed Riemannian manifolds of positive Ricci curvature
I will demonstrate that on any closed Riemannian manifold of dimension n with Ricci curvature greater
than or equal to n-1, there exists a closed geodesic of length at most 8 \pi n.
Melanie Rupflin
Lojasiewicz inequalities near simple bubble trees for geometric variational problems
In the study of (almost) critical points of a geometric variational problem one is often confronted
with the problem that a weakly-obtained limiting object does not have the same topology. A major
challenge in such situations is that the seminal results of Simon on Lojasiewicz inequalities, one of
the most powerful tools in the analysis of the energy spectrum of analytic energies and the
corresponding gradient flows, are not applicable. In this talk we present a method that allows us to
prove such Lojasiewicz inequalities also when the topology changes e.g. due to the formation of a
bubble and present applications for the H-surface energy and the harmonic map heat flow.
Catherine Searle
Almost isotropy-maximal manifolds of non-negative curvature
We extend the equivariant classification results of Escher and Searle for closed, simply connected,
non-negatively curved Riemannian n-manifolds admitting isometric isotropy-maximal torus actions
to the class of such manifolds admitting isometric strictly almost isotropy-maximal torus actions. In
particular, we prove that such manifolds are equivariantly diffeomorphic to the free, linear quotient
by a torus of a product of spheres of dimensions greater than or equal to three. This is joint work
with C. Escher and Z. Dong.
Valentino Tosatti
Immortal solutions of the Kähler-Ricci flow
I will discuss what is known, not known, and conjectured about solutions of the Kähler-Ricci flow on
compact Kähler manifolds which exist for all positive time.
Thomas Walpuski
Genus bounds for pseudo-holomorphic curves
Pseudo-holomorphic curves in symplectic manifolds with compatible almost complex structures are a fascinating class of (possibly singular) minimal surfaces. They have found numerous applications in symplectic geometry and share many features of holomorphic curves in complex projective manifolds. A very classical result due to Castelnuovo states that the genus of a holomorphic curve in complex projective space can be bounded a priori in terms of the degree of the curve and the dimension of the ambient space. The motivating question for this talk is: do analogues Castelnuovo’s genus bound hold in symplectic geometry?
In dimension four, McDuff proved that Castelnuovo's genus bound carries over to the symplectic setting. In higher dimensions, Gromov's h-principle forbids an unconditional analogue of Castelnuovo's genus bound independent of the choice of almost complex structure. However, in dimension at least eight, for generic almost complex structures very strong genus bounds can be derived from the index theorem. This leaves the case of symplectic 6-manifolds; in particular, symplectic Calabi–Yau 3-folds.
By combining recent breakthroughs in geometric measure theory by De Lellis–Spadaro–Spolaor and equivariant transversality by Wendl a non-effective generic genus bound (depending on the choice of almost complex structure) for symplectic Calabi–Yau 3-folds can be derived. To overcome the dependence on the choice of almost complex structure, one can reformulate the question to consider non-zero contributions to curve counting invariants in a certain genus (instead of individual curves). This leads to the Gopakumar–Vafa finiteness conjecture. This too can be resolved using tools from GMT combined with Ionel–Parker's concept of clusters.
This is joint work with Aleksander Doan and Eleny Ionel.
Guofang Wei
Fundamental gap estimate in hyperbolic space
In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap conjecture that
difference of first two eigenvalues of the Laplacian with Dirichlet boundary condition on convex domain
with diameter D in the Euclidean space is greater than or equal to $3\pi^2/D^2$. In several joint works
with X. Dai, Z. He, S. Seto, L. Wang (in various subsets) the estimate is generalized, showing the same
lower bound holds for convex domains in the unit sphere. In sharp contrast, in recent joint work with T.
Bourni, J. Clutterbuck, X. Nguyen, A. Stancu and V. Wheeler, we prove that the product of the
fundamental gap with the square of the diameter can be arbitrarily small for convex domains of any
diameter in hyperbolic space. Very recently, joint with X. Nguyen, A. Stancu, we show even for
horoconvex domains in the hyperbolic space, the product of their fundamental gap with the square of
their diameter has no positive lower bound.
