Zoom Link

In this talk, we will discuss some of Gromov's conjectures on scalar curvature from the perspective of general relativity. 

Zoom Link

Frieder Jäckel: The pressure metric on quasi-Fuchsian space

Abstract: Quasi-Fuchsian space is a moduli space of certain hyperbolic 3-manifolds and it can be equipped with a metric, called the pressure metric, whose definition is inspired by the definition of the Weil-Petersson metric on Teichmüller space. We will discuss that the pressure metric on quasi-Fuchsian space has finite diameter, why this is interesting, and how it relates to the study of eigenfunctions. The talk is based on joint work with E. Fioravanti, U. Hamenstädt and Y. Zhang

Yifan Guo: Green's functions on minimal submanifolds

Abstract: We are going to discuss the following properties of Green's function on a minimal submanifold with singularities : lower and upper bounds, asymptotics near the pole and infinity as well as $C^{\alpha}$ convergence under convergence of minimal submanifolds. We show that the Green's functions on area-minimizing boundaries have all of the mentioned properties. We also have $L^p$ convergence of the Green's functions for converging multiplicity 1 stationary varifolds. Outside of these classes of minimal submanifolds, the properties may fail and we discuss the reasons and some examples.

Zoom Link

Free boundary minimal surfaces are critical points to the area functional with respect to boundary-preserving deformations. Apart from being a natural Neumann boundary value problem for minimal surfaces, these objects also arise in many other geometric problems like the extremal Steklov eigenvalue problem studied in the celebrated work of Fraser and Schoen. In this talk, I will discuss the question of existence and regularity for free boundary minimal surfaces, focusing in particular on certain min-max constructions where tools from geometric measure theory play a major role.

Zoom Link

G_2-instantons are a special kind of connections on a Riemannian 7-manifold,  analogues of anti-self-dual connections in 4 dimensions.

I will start this talk by introducing G_2-instantons and their significance, and then describe new families of SU(2)^2-invariant G_2-instantons in cohomogeneity one manifolds with a coclosed G_2-structure.

Zoom Link

The avoidance principle says that two mean curvature flows of hypersurfaces remain disjoint if they are disjoint initially. In this talk, we will discuss several generalizations of the avoidance principle that allow for intersections between the two hypersurfaces. First, we show that the Hausdorff dimension of the intersection of two mean curvature flows is non-increasing over time. We then prove a dimension monotonicity result for self-intersections of immersed mean curvature flows. Next, we extend the intersection dimension monotonicity to a class of Brakke flows and level set flows, and we provide examples showing that this monotonicity fails for general Brakke flows. In the course of the proof, we obtain a localization result and an equivalence between non-fattening and non-discrepancy for level set flows with finitely many singularities. This is joint work with Tang-Kai Lee.

Gabor, Section 3

P1 Prop 15 & Thm 16

The purpose of these series of lectures is to explain a recent result of Gabor Székelyhidi [Szek24] ttps://arxiv.org/abs/2408.10747, which shows that many singular Kähler-Einstein (K-E) spaces are quasi-smooth non collapsed RCD metric spaces with interesting geometric features.  

We shall focus on Section 3 of [Szek24], showing that the metric completion of a constant scalar curvature Kähler (cscK) approximable K-E space is a non collapsed quasi-smooth RCD space, and that the corresponding singular K-E metric is a Kähler current.

Gabor, Section 4

The purpose of these series of lectures is to explain a recent result of Gabor Székelyhidi [Szek24] ttps://arxiv.org/abs/2408.10747, which shows that many singular Kähler-Einstein (K-E) spaces are quasi-smooth non collapsed RCD metric spaces with interesting geometric features. 

We shall focus on Section 4 of [Szek24], showing that metric completion of a cscK approximable K-E space is homeomorphic to the analytic completion, and that the singular metric set coincides with the analytic singular set, in particular it has codimension greater than 4.

Siobhan Roberts will recount her trajectory from history major to math journalist, and give a behind-the-scenes look at writing about mathematics for the New York Times.

Zoom Link

The purpose of these series of lectures is to explain a recent result of Gabor Székelyhidi [Szek24] ttps://arxiv.org/abs/2408.10747, which shows that many singular Kähler-Einstein (K-E) spaces are quasi-smooth non collapsed RCD metric spaces with interesting geometric features. 

We shall focus on Section 4 of [Szek24], showing that metric completion of a cscK approximable K-E space is homeomorphic to the analytic completion, and that the singular metric set coincides with the analytic singular set, in particular it has codimension greater than 4.

Gabor, Section 5

The purpose of these series of lectures is to explain a recent result of Gabor Székelyhidi [Szek24] ttps://arxiv.org/abs/2408.10747, which shows that many singular Kähler-Einstein (K-E) spaces are quasi-smooth non collapsed RCD metric spaces with interesting geometric features. 

We will provide several examples of K-E spaces that are cscK approximable.

Zoom Link

Gabor Section 2, Cor 8

The purpose of these series of lectures is to explain a recent result of Gabor Székelyhidi [Szek24] ttps://arxiv.org/abs/2408.10747, which shows that many singular Kähler-Einstein (K-E) spaces are quasi-smooth non collapsed RCD metric spaces with interesting geometric features.  

These lectures will provide a brief introduction to RCD spaces, explaining the background needed for [Szek24], as well as some interesting geometric consequences of this property.

Taught courses are metric cones and stratification.

The purpose of these series of lectures is to explain a recent result of Gabor Székelyhidi [Szek24] ttps://arxiv.org/abs/2408.10747, which shows that many singular Kähler-Einstein (K-E) spaces are quasi-smooth non collapsed RCD metric spaces with interesting geometric features.  

These lectures will provide a brief introduction to RCD spaces, explaining the background needed for [Szek24], as well as some interesting geometric consequences of this property.

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link