Abstracts (in alphabetical order)
Christian Bär
Rigidity results for scalar curvature
The talk will consist of two parts. In the first one, a relation between the hyperspherical radius and Dirac eigenvalues will be used to give simple proofs of some classical rigidity results for scalar curvature and to establish a new relation between the hyperspherical radius and the Yamabe invariant. In the second part, the condition in Llarull's theorem that the comparison map has nonzero degree will be discussed.
Timothy Buttsworth
Computationally-assisted existence proofs in differential geometry
In this talk, I will outline a general two-step approach that has the potential to be very useful in the construction of many ad hoc examples of special solutions of geometric PDEs. The first step is to obtain a (verifiably) highly-accurate approximate solution metric using numerical techniques, and the second step is to then use perturbative techniques to prove existence of a true solution near the approximate solution. I will report on recent work I have done with Liam Hodgkinson (Melbourne) which successfully uses this approach to prove existence of a new O(3)×O(10)-invariant Einstein metric on S^{12}. I will then go on to discuss other possible geometric applications.
Lorenzo Foscolo
Unstable minimal surfaces in hyperkähler 4-manifolds
I will describe the construction of new minimal surfaces in hyperkähler 4-manifolds arising from the Gibbons–Hawking Ansatz, i.e. hyperkähler 4-manifolds that admit a triholomorphic circle action, and on certain K3 surfaces. The minimal surfaces we produce are obtained via a gluing construction using well-known minimal surfaces, the Scherk surface in flat space and the holomorphic cigar in the Taub-NUT space, as building blocks. The minimal surfaces obtained via this construction are not holomorphic with respect to any complex structure compatible with the metric, are not circle invariant, they can be parameterized by a harmonic map that satisfies a first-order Fueter-type PDE, and yet are unstable. This shows that there is no characterisation of stable minimal surfaces in hyperkähler 4-manifolds in terms of topological data. This is joint work with Federico Trinca.
Ursula Hamenstädt
Negatively curved Kaehler Einstein metrics not covered by the ball
For any $n\geq 2$ we construct a family of closed complex manifolds of dimension n which admit a Kaehler Einstein metric of negative sectional curvature, but whose universal covering is not biholomorphic to the ball. This is joint work with Henri Guenanica.
Lee Kennard
Positive intermediate Ricci curvature and Hopf’s conjecture
Hopf conjectured that even-dimensional closed Riemannian manifolds with positive sectional curvature have positive Euler characteristic. The conclusion is known to fail if the curvature assumption is relaxed in any number of ways, including to positive intermediate Ricci curvature and even in the presence of symmetry. This talk will present a positive result for positive second intermediate Ricci curvature that requires torus symmetry whose rank is independent of the manifold dimension. Multiple extensions of obstructions to positive sectional curvature are required, but the main new one is a non-trivial extension of the speaker’s Four Periodicity Theorem by Nienhaus’s 2018 M.Sc. thesis to the case of partially periodic cohomology rings. This is joint work with Lawrence Mouillé and Jan Nienhaus.
Man Chun Lee
Existence and uniqueness of Ricci flow smoothing
In this talk, we will discuss the existence and uniqueness of Ricci flow smoothing on complete non-compact manifolds without bounded curvature conditions. We will discuss some known related uniqueness and their applications to geometric problems.
John Lott
Long time behavior of Ricci flow on some complex surfaces
I'll give biLipschitz models for the Ricci flow on some 4-manifolds (minimal surfaces of general type) that exhibit a combination of expanding and static behavior.
Ursula Ludwig
Torsion and refined torsion on singular spaces
The comparison between analytic and topological torsion of a smooth compact manifold equipped with a unitary flat vector bundle, aka Cheeger-Müller theorem, is one of the most important comparison theorems in global analysis. Refined versions of both analytic and topological torsion on a smooth compact manifold have been amply studied as well. The aim of this talk is to present extensions of the Cheeger-Müller theorem for torsion as well as for refined torsion on singular spaces with conical singularities.
Alexander Lytchak
Smoothness of Riemannian submersions and submetries
In the talk I will explain a proof of the following statement and some generalizations thereof, obtained jointly with Burkhard Wilking: A Riemannian submersion between positively curved smooth Riemannian manifolds is always a smooth map.
Elena Mäder-Baumdicker
The Willmore energy landscape below $12\pi$
In certain contexts, the Willmore energy can be interpreted as a measure of complexity for closed surfaces. One example is its role in theory of the sphere eversion -- constructing a smooth path through immersed spheres from the round sphere to its inside-out counterpart. In this talk, I present new results concerning the Willmore energy landscape for immersed spheres with energy below $12\pi$.We show that this space decomposes into four distinct regular homotopy classes. Two of these classes do not contain any round sphere, and initiating the Willmore flow from a surface in one of them inevitably leads to a singularity.
For the remaining two classes, we construct regular homotopies to the round sphere such that the Willmore energy never exceeds that of the initial surface. This talk is based on joint work with Jona Seidel.
Lei Ni
Compact complex homogeneous manifolds without derivatives
With N. Wallach, for a given compact Lie algebra $\mathfrak{g}$, a subalgebra $\mathfrak{k}$, and an invariant almost complex structure $J$, we give a characterization when $J$ is integrable, and associate $J$ with a canonical Cartan subalgebra of $\mathfrak{g}$. This extends (implies) classic results of H.-C. Wang, J. Tits, Wolf-Wang-Ziller on compact complex homogeneous manifolds.
Catherine Searle
Almost non-negatively curved 5-manifolds with torus symmetry
We generalize the almost maximal symmetry rank result for closed, simply connected, non-negatively curved 5-manifolds of Galaz-García and Searle to almost non-negative curvature. In particular, we show that such manifolds are diffeomorphic to $S^5$, $S^3\times S^2$, $S^3\tilde{\times}S^3$, the non-trivial $S^3$ bundle over $S^2$, or the Wu manifold, $SU(3)/SO(3)$. This is joint work with Samuel Bartel and John Harvey.
Uwe Semmelmann
Quaternion Kähler manifolds of non-negative sectional curvature
Quaternion Kähler manifolds, i.e., Riemannian manifolds with holonomy contained in Sp(m)Sp(1), are Einstein. In the case of positive scalar curvature, there is a longstanding conjecture by LeBrun and Salamon stating that all such manifolds should be symmetric. So far, the conjecture has been confirmed only up to dimension 12.
In my talk, I will present a proof of the conjecture under the additional assumption of non-negative sectional curvature. This extends earlier work by Berger, who proved that quaternion Kähler manifolds of positive sectional curvature are isometric to the quaternionic projective space. My talk is based on a joint article with Simon Brendle and on earlier work by Simon Brendle.
Mark Stern
The geometry of p-harmonic forms
In this talk we describe basic geometric properties of p-harmonic forms and p-coclosed forms and use them to reprove vanishing theorems of Pansu and new injectivity theorems for the Lp -cohomology of simply connected, pinched negatively curved manifolds. We also provide a partial resolution of a conjecture of Gromov on the vanishing of Lp -cohomology on symmetric spaces.
Hunter Stufflebeam
Stability Theorems for the Width of Spheres
In this talk, I'll discuss some recent and ongoing work about the stability of min-max widths of spheres under various lower curvature bounds. Some of this is joint with Davi Maximo, and some with Paul Sweeney Jr.
Darya Sukhorebska
Positively curved 16-manifolds with symrank ≥ 3
This is a joint work with Burkhard Wilking. We investigate positively curved orientable 16-manifolds with a torus rank 3 and we show that their Euler characteristic coincides with one of the 16-dimensional rank one symmetric spaces. This statement on Euler characteristic was previously proved by Manuel Amann and Lee Kennard for the manifolds with the symmetry rank 4. One of the main tools in our case is to examine the graph associated to the corresponding Z_2^3-subaction. This method was inspired by the work of Lee Kennard, Michael Wiemeler and Burkhard Wilking. However, we introduce a slightly modified version, that seems to be better adapted for low dimensions.
Adam Thompson
Harmonic-Einstein metrics on Riemann surfaces and Higgs bundles
I will talk about the harmonic-Einstein equation, a coupled system of equations for a harmonic map between Riemannian manifolds and a Einstein-like metric on the domain. This system is of independent interest, but also arises in the study of Ricci solitons (and Einstein metrics) with symmmetry. I will focus on the case the domain is a Riemann surface and explain a link with Higgs bundles. This is based on joint work with R. Lafuente.
Gregor Weingart
Stable curvature invariant
Integrating local curvature invariants over of compact Riemannian manifolds produces a set of global invariants of Riemannian manifolds,
which can be used for example to distinguish different isometry classes of such manifolds. The aim of this talk is to provide a graphical calculus akin to the calculus of Rozansky--Witten invariants to describe the polynomial invariants obtained in this way, which are stable in the sense that they are defined in arbitrary dimensions. Among the characteristic numbers of a Riemannian manifold the Euler characteristic is the only stable curvature invariant in this sense, we will describe it completely in the emerging graphical calculus together with the strikingly similar generating function of the moments of sectional curvature. Moreover we will use the graphical calculus to derive the cubic analogue of the quadratic Hitchin--Thorpe identity for Einstein manifolds of arbitrary dimensions.
David Wraith
Anand Dessai at 60
On the occasion of his 60th birthday, we take a look at the career Anand Dessai to date
Yujie Wu
Capillary hypersurfaces and variational methods in positively curved manifolds with boundaray
We study free boundary and capillary minimal hypersurfaces from the variational point of view. In particular, we study the interaction of these objects with scalar curvature and boundary convexity. We first apply the method of free boundary µ-bubbles) to study manifolds with positive scalar curvature to prove a rigidity result for free boundary minimal hypersurfaces in a 4-manifolds with certain positivity assumptions on curvature. Then we define generalized capillary surfaces (θ-bubbles) and use θ-bubbles to obtain geometric estimates on manifolds with non-negative scalar curvature and uniformly mean convex boundary, obtaining estimates and rigidity results for such manifolds.
Sergio Zamora
Non-collapsed covers and topological control
In differential geometry, a common technique to exploit a curvature bound is to study the corresponding universal cover or other suitable covering spaces. This technique is particularly useful when these covering spaces are non-collapsed--that is, the volumes of their unit balls admit a uniform positive lower bound. In this talk, we will discuss classical and recent results regarding the existence of such non-collapsed covers and related questions.
Rudolf Zeidler
Positive scalar curvature with point singularities
I will review a selection of rigidity results that have played an important role in scalar curvature geometry in recent years. I will place particular emphasis on problems involving Riemannian metrics with point singularities. This includes a topological construction of positive scalar curvature (psc) metrics with uniformly Euclidean ($L^\infty$) point singularities on manifolds that do not admit smooth psc metrics. Based on joint work with Simone Cecchini and Georg Frenck.
Shengxuan Zhou
Examples related to Ricci limit spaces and topology
In this talk, we will describe the construction of two examples related to Ricci limit spaces:
(1). For any n≥3, there exists an n-dimensional Ricci limit space has no open subset which is topologically a manifold. This generalizes a result of Hupp-Naber-Wang. As a corollary, our example provides a collapsed sequence of boundary free manifolds whose limit has a dense boundary with infinitely many connected components.
(2). For any n≥4, there exists a sequence of n-dimensional tori with Ricci lower bound that converges to a singular space. This answers a question posted by Petrunin and Brue-Naber-Semola. In the 4-dimensional case, we prove that the Gromov-Hausdorff limit of tori with a two-sided Ricci bound and a diameter bound is always a topological torus.
