Panagiota Daskalopoulos: Ancient solutions to geometric flows

Some of the most important problems in PDE are related to the understanding of singularities. This usually happens through a blow up procedure near the potential singularity which uses the scaling properties of the equation. In the case of a parabolic equation the blow up analysis often leads to special solutions which are defined for all time \(- \infty < t \leq T\), for some \(T \leq +\infty\). We refer to them as ancient solutions.

In this lecture we will discuss Uniqueness Theorems for ancient compact solutions to the Ricci flow and mean curvature flow. 

Jingyin Huang: The Helly geometry of some fundamental groups of complex hyperplane arrangement complements

A complex hyperplane complement is a topological space obtained by removing a collection of complex codimension one affine hyperplanes from \(\mathbb{C}^n\) (or a convex cone of \(\mathbb{C}^n\)). Despite the simple definition, these spaces have highly non-trivial topology. They naturally emerge from the study of real and complex reflection groups, braid groups and configuration spaces, and Artin groups. More recently, the fundamental groups of some of these spaces start to play important roles in geometric group theory, though most of these groups remain rather mysterious. We introduce a geometric way to understand classes of fundamental groups of some of these spaces, by equivariantly “thickening” these groups to metric spaces which satisfy a specific geometric property that is closely related to convexity and non-positive curvature. We also discuss several algorithmic, geometric and topological consequences of such a non-positive curvature condition. This is joint work with D. Osajda.

Wenshuai Jiang: Gromov–Hausdorff limit of manifolds and some applications

The Gromov–Hausdorff distance is a distance between two metric spaces, which was introduced by Gromov in 1981. From Gromov’s compactness theorem, we knew that any sequence of manifolds with uniform lower Ricci curvature bounds has a converging subsequence in Gromov–Hausdorff topology to a limit metric space. The limit metric space in general may not be a manifold. The singularity and regularity of such limit metric space has been studied widely since 1990. It turns out that such study has powerful applications in geometry. In fact, the resolution of the Yau–Tian–Donaldson conjecture was largely relied on the development of the study to the limit metric space.

In the first part of the talk, we will discuss some recent progress of the Gromov–Hausdorff limit of a sequence of manifolds with Ricci curvature bounds. In the second part, we will discuss some applications.

Chao Li: The geometry and topology of scalar curvature in low dimensions

Scalar curvature is a weak curvature invariant of a Riemannian metric. An interesting question is which topological/geometric conditions obstruct the existence of metrics with positive scalar curvature. In particular, a well-known conjecture states that a closed aspherical manifold does not admit any Riemannian metric with positive scalar curvature.

In this talk, we will describe joint work with Otis Chodosh including a proof of this conjecture in 4 and 5 dimensions. In addition, we will briefly describe how the techniques may be applied to related problems, e.g. rigidity of minimal hypersurfaces in positively curved 4–manifolds.

Ciprian Manolescu: A knot Floer stable homotopy type

Knot Floer homology (introduced by Ozsváth–Szabó and Rasmussen) is an invariant whose definition is based on symplectic geometry, and whose applications have transformed knot theory over the last two decades. Starting from a grid diagram of a knot, I will explain how to construct a spectrum whose homology is knot Floer homology. Conjecturally, the homotopy type of the spectrum is an invariant of the knot. The construction does not use symplectic geometry, but rather builds on a combinatorial definition of knot Floer homology. We inductively define models for the moduli spaces of pseudo-holomorphic strips and disk bubbles, and patch them together into a framed flow category. The inductive step relies on the vanishing of an obstruction class that takes values in a complex of positive domains with partitions. (This is joint work with Sucharit Sarkar.)

Assaf Naor: Extension, separation and isomorphic reverse isoperimetry

In the Lipschitz extension problem we are given a pair of metric spaces \(X\), \(Y\) and ask for the smallest \(K\) such that for any subset \(A\) of \(X\) every \(L\)–Lipschitz mapping from \(A\) to \(Y\) can be extended to a \(KL\)–Lipschitz mapping from \(X\) to \(Y\). We will describe progress on Lipschitz extension that relates it to well-studied volumetric issues in convex geometry and to a new question on reversing the isoperimetric inequality.

André Neves: Geodesics and minimal surfaces

There are several properties of closed geodesics which are proven using its Hamiltonian formulation (which has no analogue for minimal surfaces). I will talk about some recent progress in proving some of these properties for minimal surfaces. 

Lu Wang: Hypersurfaces of low entropy are isotopically trivial

Colding and Minicozzi define the entropy of a hypersurface to be the supremum of Gaussian integrals with varying centers and scales. In this talk, I will discuss recent results on the relationship between topology and entropy of hypersurfaces. 

Ruobing Zhang: Metric geometry of Calabi–Yau manifolds in complex dimension two

This talk focuses on the recent resolution of three folklore conjectures in the field (joint with Song Sun). 

1. Any volume collapsed limit of unit-diameter Calabi–Yau metrics on the K3 manifold is isometric to one of the following: a flat 3–torus quotient by revolution, a special Kähler metric on a 2–sphere, and the unit interval.
2. Any gravitational instanton, as a bubble limit, has one of the six asymptotic model geometries (with optimal asymptotic rates): ALE, ALF, ALG, ALG*, ALH, and ALH*.
3. Any gravitational instanton can be holomorphically compactified to be an open dense subset of a compact algebraic surface. 

With the above classification results, we obtain a rather complete picture of degenerating Calabi–Yau manifolds in complex dimension two.