# Geometry and Topology Seminar

The Geometry and Topology seminar meets in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm.

## Fall 2017

date speaker title host(s)
September 8 TBA TBA TBA
September 15 Jiyuan Han (University of Wisconsin-Madison) "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces" Local
September 22 Sigurd Angenent (UW-Madison) "Topology of closed geodesics on surfaces and curve shortening" Local
September 29 Ke Zhu (Minnesota State University) "Isometric Embedding via Heat Kernel" Bing Wang
October 6 Shaosai Huang (Stony Brook) "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons" Bing Wang
October 13 Sebastian Baader (Bern) "A filtration of the Gordian complex via symmetric groups" Kjuchukova
October 20 Shengwen Wang (Johns Hopkins) "Hausdorff stability of round spheres under small-entropy perturbation" Lu Wang
October 27 Marco Mendez-Guaraco (Chicago) TBA Lu Wang
November 3 TBA TBA TBA
November 10 TBA TBA TBA
November 17 Ovidiu Munteanu (University of Connecticut) "The geometry of four dimensional shrinking Ricci solitons" Bing Wang
Thanksgiving Recess
December 1 TBA TBA TBA
December 8 Brian Hepler (Northeastern University) "Perverse Results on Parameterized Hypersurfaces" Max

## Fall Abstracts

### Jiyuan Han

"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"

Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with Jeff Viaclovsky.

### Sigurd Angenent

"Topology of closed geodesics on surfaces and curve shortening"

A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.

### Ke Zhu

"Isometric Embedding via Heat Kernel"

The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.

### Shaosai Huang

"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"

A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local L^{\infty}-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as \epsilon-regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an \epsilon-regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant.

"A filtration of the Gordian complex via symmetric groups"

The Gordian complex is a countable graph whose vertices correspond to knot types and whose edges correspond to pairs of knots that are related by a crossing change in a suitable diagram. For every natural number n, we consider the subgraph of the Gordian complex defined by restricting to the knot types whose fundamental group surjects onto S_n. We will prove that the various inclusion maps from these subgraphs into the Gordian complex are isometric embeddings. From this, we obtain a simple metric filtration of the Gordian complex.

### Shengwen Wang

"Hausdorff stability of round spheres under small-entropy perturbation"

Colding-Minicozzi introduced the entropy functional on the space of all hypersurfaces in the Euclidean space when studying generic singularities of mean curvature flow. It is a measure of complexity of hypersurfaces. Bernstein-Wang proved that round n-spheres minimize entropy among all closed hypersurfaces for n less than or equal to 6, and the result is generalized to all dimensions by Zhu. Bernstein-Wang later also proved that the round 2-sphere is actually Hausdorff stable under small-entropy perturbations. I will present in this talk the generalization of the Hausdorff stability to round hyper-spheres in all dimensions.

"TBA"

### Ovidiu Munteanu

"The geometry of four dimensional shrinking Ricci solitons"

I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons.


### Brian Hepler

"Perverse Results on Parameterized Hypersurfaces"

We discuss some results for the cohomology of Milnor fibers inside parameterized hypersurfaces which follow quickly from results in the category of perverse sheaves. In particular, we define a new perverse sheaf called the multiple-point complex of the parameterization, which naturally arises when investigating how the multiple-point set influences the topology of the Milnor fiber. Time Permitting, we will discuss applications to one-parameter deformations of such hypersurfaces. This is joint work with David Massey.