Difference between revisions of "Geometry and Topology Seminar"

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The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.
 
The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.
 
<br>  
 
<br>  
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .
+
For more information, contact Shaosai Huang.
  
 
[[Image:Hawk.jpg|thumb|300px]]
 
[[Image:Hawk.jpg|thumb|300px]]
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== Fall 2017 ==
+
 
 +
== Fall 2019 ==
  
 
{| cellpadding="8"
 
{| cellpadding="8"
Line 15: Line 16:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|September 8
+
|Oct. 4
|TBA
+
|Ruobing Zhang (Stony Brook University)
|TBA
+
| Geometric analysis of collapsing Calabi-Yau spaces
|TBA
+
|(Chen)
|-
 
|September 15
 
|Jiyuan Han (University of Wisconsin-Madison)
 
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]
 
|Local
 
|-
 
|September 22
 
|Sigurd Angenent (UW-Madison)
 
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]
 
|Local
 
|-
 
|September 29
 
|Ke Zhu (Minnesota State University)
 
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]
 
|Bing Wang
 
|-
 
|October 6
 
|Shaosai Huang (Stony Brook)
 
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]
 
|Bing Wang
 
|-
 
|October 13
 
|Sebastian Baader (Bern)
 
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]
 
|Kjuchukova
 
 
|-
 
|-
|October 20
 
|Shengwen Wang (Johns Hopkins)
 
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]
 
|Lu Wang
 
 
|-
 
|-
|October 27
+
|Oct. 25
|Marco Mendez-Guaraco (Chicago)
+
|Emily Stark (Utah)
|TBA
+
| Action rigidity for free products of hyperbolic manifold groups
|Lu Wang
+
|(Dymarz)
 
|-
 
|-
|November 3
+
|Nov. 8
|TBA
+
|Max Forester (University of Oklahoma)
|TBA
+
|Spectral gaps for stable commutator length in some cubulated groups
|TBA
+
|(Dymarz)
 
|-
 
|-
|November 10
+
|Nov. 22
|TBA
+
|Yu Li (Stony Brook University)
|TBA
+
|On the structure of Ricci shrinkers
|TBA
+
|(Huang)
|-
 
|November 17
 
|Ovidiu Munteanu (University of Connecticut)
 
|[[#Ovidiu Munteanu| "The geometry of four dimensional shrinking Ricci solitons"]]
 
|Bing Wang
 
|-
 
|<b>Thanksgiving Recess</b>
 
|
 
|
 
|
 
|-
 
|December 1
 
|TBA
 
|TBA
 
|TBA
 
|-
 
|December 8
 
|Brian Hepler (Northeastern University)
 
|[[#Brian Hepler| "Perverse Results on Parameterized Hypersurfaces"]]
 
|Max
 
 
|-
 
|-
 
|}
 
|}
  
== Fall Abstracts ==
+
==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.
+
===Ruobing Zhang===
  
=== Shengwen Wang ===
+
This talk centers on the degenerations of Calabi-Yau metrics. We will focus on the interactions between algebraic degenerations and metric convergence with highly singular behaviors in the collapsing case. As the complex structures degenerate, the collapsing Calabi-Yau metrics may exhibit various wild geometric properties with highly non-algebraic features.
"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.
+
First, as motivating examples, we will describe our recent results on the new collapsing mechanisms of K3 surfaces. Next, we will switch to higher dimensions and we will exhibit some entirely new constructions of degenerating Calabi-Yau metrics which are expected to work in broader contexts. Complex structures degeneration will be accurately characterized by the bubbling and singularity analysis in a geometric manner.
  
=== Marco Mendez-Guaraco ===
+
===Emily Stark===
"TBA"
 
  
=== Ovidiu Munteanu ===
+
The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold group is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.
"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.
+
===Max Forester===
  
=== Brian Hepler ===
+
I will discuss stable commutator length (scl) in groups, and some gap theorems for the scl spectrum. Such results say that for various groups, scl of an element is always either zero or is larger than some uniform constant. I will discuss the cases of right-angled Artin groups and certain right-angled Coxeter groups. This is joint work with Pallavi Dani, Ignat Soroko, and Jing Tao.
"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.
+
===Yu Li===
 +
We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As an application, we show that any Ricci shrinker whose second eigenvalue of the curvature operator is positive must be a quotient of sphere.
  
 
== Archive of past Geometry seminars ==
 
== Archive of past Geometry seminars ==
 +
2018-2019  [[Geometry_and_Topology_Seminar_2018-2019]]
 +
<br><br>
 +
2017-2018 [[Geometry_and_Topology_Seminar_2017-2018]]
 +
<br><br>
 
2016-2017  [[Geometry_and_Topology_Seminar_2016-2017]]
 
2016-2017  [[Geometry_and_Topology_Seminar_2016-2017]]
 
<br><br>
 
<br><br>

Latest revision as of 08:44, 4 November 2019

The Geometry and Topology seminar meets in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm.
For more information, contact Shaosai Huang.

Hawk.jpg



Fall 2019

date speaker title host(s)
Oct. 4 Ruobing Zhang (Stony Brook University) Geometric analysis of collapsing Calabi-Yau spaces (Chen)
Oct. 25 Emily Stark (Utah) Action rigidity for free products of hyperbolic manifold groups (Dymarz)
Nov. 8 Max Forester (University of Oklahoma) Spectral gaps for stable commutator length in some cubulated groups (Dymarz)
Nov. 22 Yu Li (Stony Brook University) On the structure of Ricci shrinkers (Huang)

Fall Abstracts

Ruobing Zhang

This talk centers on the degenerations of Calabi-Yau metrics. We will focus on the interactions between algebraic degenerations and metric convergence with highly singular behaviors in the collapsing case. As the complex structures degenerate, the collapsing Calabi-Yau metrics may exhibit various wild geometric properties with highly non-algebraic features.

First, as motivating examples, we will describe our recent results on the new collapsing mechanisms of K3 surfaces. Next, we will switch to higher dimensions and we will exhibit some entirely new constructions of degenerating Calabi-Yau metrics which are expected to work in broader contexts. Complex structures degeneration will be accurately characterized by the bubbling and singularity analysis in a geometric manner.

Emily Stark

The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold group is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.

Max Forester

I will discuss stable commutator length (scl) in groups, and some gap theorems for the scl spectrum. Such results say that for various groups, scl of an element is always either zero or is larger than some uniform constant. I will discuss the cases of right-angled Artin groups and certain right-angled Coxeter groups. This is joint work with Pallavi Dani, Ignat Soroko, and Jing Tao.

Yu Li

We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As an application, we show that any Ricci shrinker whose second eigenvalue of the curvature operator is positive must be a quotient of sphere.

Archive of past Geometry seminars

2018-2019 Geometry_and_Topology_Seminar_2018-2019

2017-2018 Geometry_and_Topology_Seminar_2017-2018

2016-2017 Geometry_and_Topology_Seminar_2016-2017

2015-2016: Geometry_and_Topology_Seminar_2015-2016

2014-2015: Geometry_and_Topology_Seminar_2014-2015

2013-2014: Geometry_and_Topology_Seminar_2013-2014

2012-2013: Geometry_and_Topology_Seminar_2012-2013

2011-2012: Geometry_and_Topology_Seminar_2011-2012

2010: Fall-2010-Geometry-Topology