Geometry and Topology Seminar 2022 2023

Fall 2022

date	speaker	title
Sept. 23	Ruobing Zhang (Princeton)	Metric geometry of hyperkähler four-manifolds
Oct. 14	Min Ru (University of Houston) (joint w/ Analysis seminar)	The K-stability and Nevanlinna/Diophantine theory
Nov. 4	Jesse Madnick (University of Oregon)	Cohomogeneity-One Lagrangian Mean Curvature Flow
Nov. 11	Gavin Ball	Associative submanifolds of some nearly parallel G2-manifolds

Fall abstracts

Ruobing Zhang

This talk focuses on the recent resolution of the following three well-known conjectures in the field.

(1) Any volume collapsed limit of unit-diameter K3 metrics is isometrically classified as: the quotient of a flat 3D torus by an involution, a singular special Kähler metric on the topological 2-sphere, or the unit interval.

(2) Any complete non-compact hyperkähler 4-manifold with quadratically integrable curvature, must have a classified model end.

(3) Any gravitational instanton can be compactified to an open dense subset of certain compact algebraic surface.

Therefore, in the hyperkähler setting, we obtain a rather complete picture of the metric geometry on all scales.

Min Ru

In the recent paper with P. Vojta, we introduced the so-called beta-constant, and used it to extend the Cartan's Second Main Theorem in Nevanlinna theory and Schmidt's subspace theorem in Diophantine approximation. It turns out the beta-constant is also used in the algebro-geometric stability criterion in the Fano's case. In this talk, I'll describe and explore the somewhat mysterious connection. The talk is based on the recent joint paper with Yan He entitled "The stability threshold and Diophantine approximation", Proc. AMS, 2022.

Jesse Madnick

In complex n-space, mean curvature flow preserves the class of Lagrangian submanifolds, a fact known as “Lagrangian mean curvature flow” (LMCF). As LMCF typically forms finite-time singularities, it is of interest to understand the blowup models of such singularities, as well as the possible soliton solutions.

In this talk, we'll consider the mean curvature flow of Lagrangians that are cohomogeneity-one under the action of a compact Lie group. Interestingly, each such Lagrangian lies in a level set \mu^{-1}(\xi) of the moment map, and mean curvature flow preserves this containment. Using this fact, we'll classify all cohomogeneity-one shrinking, expanding, and translating solitons. Further, in the zero level set \mu^{-1}(0), we'll classify the Type I and Type II blowup models of cohomogeneity-one LMCF singularities. Finally, given any cohomogeneity-one special Lagrangian in \mu^{-1}(0), we'll show that it arises as the Type II blowup model of an LMCF singularity, thereby yielding infinitely many new singularity models. This is joint work with Albert Wood.

Gavin Ball

A nearly parallel G2-structure is the natural geometric structure induced on the seven-dimensional link of a conical manifold with holonomy Spin(7). The link of a conical Cayley submanifold gives an associative submanifold of the nearly parallel G2-manifold, and thus associative submanifolds of nearly parallel G2-manifolds provide models for conically singular Cayley submanifolds. In my talk I will give an introduction to nearly parallel G2-manifolds and associative manifolds and, if time permits, explain a construction of certain associative submanifolds in two settings: in the Berger space SO(5)/SO(3) with its homogenous nearly parallel G2-structure, and in squashed 3-Sasakian manifolds. In both cases the submanifolds are ruled by a special class of geodesics and arise from a construction based on holomorphic curves in the spaces of rulings. This is joint work with Jesse Madnick.

Spring 2023

Ruobing Zhang Minicourse: Topics in Metric Riemannian Geometry

Mon May 01: 2:25 pm - 3:50 pm in Van Vleck B235

Tue May 02: 2:15 pm - 4:00 pm in Birge 348

Wed May 03: 10:00 am - 11:45 am in Van Vleck B123

Thu May 04 2:15 pm - 4:00 pm in Birge 348

Fri May 05: 10:00 am - 11:45 am in Van Vleck B123

Lecture notes are available here: File:Topics in Metric Riemannian geometry.pdf