Geometry and Topology Seminar 2017-2018
The Geometry and Topology seminar meets in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm.
For more information, contact Alexandra Kjuchukova or Lu Wang .
Contents
Spring 2018
date | speaker | title | host(s) |
---|---|---|---|
January 26 | Jingrui Cheng | "Estimates for constant scalar curvature Kahler metrics with applications to existence" | Local |
February 2 | Jingrui Cheng | "Estimates for constant scalar curvature Kahler metrics with applications to existence" (continued) | Local |
February 9 | Jingrui Cheng | "Estimates for constant scalar curvature Kahler metrics with applications to existence" (continued) | Local |
February 16 | TBA | TBA | TBA |
February 23 | TBA | TBA | TBA |
March 2 | TBA | TBA | TBA |
March 9 | TBA | TBA | TBA |
March 16 | Yu Li | "The Rigidity of Ricci shrinkers of dimension four" | Bing Wang |
March 23 | TBA | TBA | TBA |
Spring Break | |||
April 6 | TBA | TBA | |
April 13 | TBA | TBA | TBA |
April 20 | Pei-Ken Hung (Columbia Univ.) | "A Minkowski inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold" | Lu Wang |
April 27 | TBA | TBA | TBA |
May 1 | Andre Neves (Distinguished Lecture) | TBA | Lu Wang |
May 2 | Andre Neves (Distinguished Lecture) | TBA | Lu Wang |
May 4 | TBA | TBA | TBA |
Spring Abstracts
Jingrui Cheng
"Estimates for constant scalar curvature Kahler metrics with applications to existence"
We develop new a priori estimates for scalar curvature type of equations on a compact Kahler manifold. As an application, we show that the properness of K-energy implies the existence of constant scalar curvature Kahler metrics. I will also talk about other applications if time permits. This is joint work with Xiuxiong Chen.
Yu Li
"The rigidity of Ricci shrinkers of dimension four"
In dimension 4, we show that a nontrivial flat cone cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature and Harnack inequality, under the pointed-Gromov- Hausdorff topology. As applications, we obtain uniform positive lower bounds of scalar curvature and potential functions on Ricci shrinkers satisfying some natural geometric properties. This is a joint work with Bing Wang.
Pei-Ken Hung
"A Minkowski inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold"
We prove a sharp inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold. This inequality generalizes the classical Minkowski inequality for surfaces in the Euclidean space, and has an interpretation from general relativity. The proof relies on a monotonicity formula for inverse mean curvature flow, and uses a geometric inequality established by Brendle.
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) | "Some geometric aspects of the Allen-Cahn equation" | 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) | "Deformation Formulas for Parameterizable 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.
Sebastian Baader
"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.
Marco Mendez-Guaraco
"Some geometric aspects of the Allen-Cahn equation"
In this talk I will discuss both local and global properties of the stationary Allen-Cahn equation in closed manifolds. This equation from the theory of phase transitions has a strong connection with the theory of minimal hypersurfaces. I will summarize recent results regarding this analogy including a new min-max proof of the celebrated Almgren-Pitts theorem.
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
"Deformation Formulas for Parameterizable Hypersurfaces"
We investigate one-parameter deformations of functions on affine space which define parameterizable hypersurfaces. With the assumption of isolated polar activity at the origin, we are able to completely express the Lê numbers of the special fiber in terms of the Lê numbers of the generic fiber and the characteristic polar multiplicities of the multiple-point complex, a perverse sheaf naturally associated to any parameterized hypersurface.