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]]
  
  
 
+
== Spring 2019 ==
== Fall 2017 ==
 
 
 
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!align="left" | date
 
!align="left" | date
Line 15: Line 13:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|September 8
+
|April  5
|TBA
+
|Mark Pengitore (Ohio)
|TBA
+
|Translation-like actions on nilpotent groups
|TBA
+
 
 +
|(Dymarz)
 
|-
 
|-
|September 15
+
|April  18
|Jiyuan Han (University of Wisconsin-Madison)
+
|José Ignacio Cogolludo Agustín (Universidad de Zaragoza)
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]
+
|Even Artin Groups, cohomological computations and other geometrical properties.
|Local
+
 
 +
|(Maxim)
 +
|'''Unusual date and time: B309 Van Vleck, 2:15-3:15'''
 
|-
 
|-
|September 22
+
 
|Sigurd Angenent (UW-Madison)
+
|April 19
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]
+
|Yan Xu (University of Missouri - St. Louis)
|Local
+
|Structure of minimal two-spheres of constant curvature in hyperquadrics
 +
|(Huang)
 +
 
 +
|}
 +
 
 +
== Fall 2018 ==
 +
 
 +
{| cellpadding="8"
 +
!align="left" | date
 +
!align="left" | speaker
 +
!align="left" | title
 +
!align="left" | host(s)
 
|-
 
|-
|September 29
+
|Sept. 14
|Ke Zhu (Minnesota State University)
+
|Teddy Einstein (UIC)
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]
+
|Quasiconvex Hierarchies for Relatively Hyperbolic Non-Positively Curved Cube Complexes
|Bing Wang
+
|(Dymarz)
 
|-
 
|-
|October 6
+
|Oct. 12
|Shaosai Huang (Stony Brook)
+
|Marissa Loving
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]
+
|Least dilatation of pure surface braids
|Bing Wang
+
|(Kent)
 
|-
 
|-
|October 13
+
|Oct. 19
|Sebastian Baader (Bern)
+
|Sara Maloni
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]
+
|On type-preserving representations of thrice punctured projective plane group
|Kjuchukova
+
|(Kent)
 
|-
 
|-
|October 20
+
|Oct. 26
|Shengwen Wang (Johns Hopkins)
+
|Dingxin Zhang (Harvard-CMSA)
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]
+
|Relative cohomology and A-hypergeometric equations
|Lu Wang
+
|(Huang)
 
|-
 
|-
|October 27
+
|Nov. 9
|Marco Mendez-Guaraco (Chicago)
+
|Zhongshan An (Stony Brook)
|TBA
+
|Ellipticity of the Bartnik Boundary Conditions
|Lu Wang
+
|(Huang)
 
|-
 
|-
|November 3
+
|Nov. 16
|TBA
+
|Xiangdong Xie
|TBA
+
|Quasi-isometric rigidity of a class of right angled Coxeter groups
|TBA
+
|(Dymarz)
|-
 
|November 10
 
|TBA
 
|TBA
 
|TBA
 
|-
 
|November 17
 
|Ovidiu Munteanu (University of Connecticut)
 
|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
 
 
|-
 
|-
 +
|
 
|}
 
|}
 +
 +
 +
==Spring Abstracts==
 +
 +
===Mark Pengitore===
 +
 +
"Translation-like actions on nilpotent groups"
 +
 +
Translation-like actions were introduced Whyte to generalize subgroup containment.  Using this notion, he proved that a group is non-amenable if and only if it admits a translation-like action by a non-abelian free group. This result motivates us to ask what groups admit translation-like actions on various interesting classes of groups. As a consequence of Gromov's polynomial growth theorem, we have that only nilpotent groups may act translation-like on a nilpotent group which is the main focus of this talk. Thus, one may ask to characterize what nilpotent groups act translation-like on a fixed nilpotent group. We offer partial answer to this question by demonstrating that if two nilpotent groups have the same growth but distinct asymptotic cones, then there exist no translation-like action of these two groups on each other.
 +
 +
===José Ignacio Cogolludo Agustín===
 +
 +
"Even Artin Groups, cohomological computations and other geometrical
 +
properties."
 +
 +
The purpose of this talk is to introduce even Artin groups and consider
 +
their quasi-projectivity properties, as well as study the cohomological
 +
properties of their kernels, that is, the kernels of their characters.
 +
 +
===Yan Xu===
 +
"Structure of minimal two-spheres of constant curvature in hyperquadrics"
 +
 +
Veronese two-sphere (also called rational normal curve) is an interesting projective variety in geometry. It is of constant curvature and unique up to action of unitary group. Based on this rigidity result and SVD (singular value decomposition) in linear algebra, we give a classification of a special class minimal, especially holomorphic, two-spheres of constant curvature in hyperquadric, up to action of real orthogonal group and reparameterization of the two-sphere. For degree less than or equal to three, we give an algorithm and explicit examples. As an application of this results, by computing the norm squared of second fundamental form, we show the generic two-spheres constructed here are not homogeneous. This is a joint work with Professor Quo-Shin Chi and Zhenxiao Xie.
  
 
== Fall Abstracts ==
 
== Fall Abstracts ==
  
=== Jiyuan Han ===
+
===Teddy Einstein===
"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
+
"Quasiconvex Hierarchies for Relatively Hyperbolic Non-Positively Curved Cube Complexes"
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 ===
+
Non-positively curved (NPC) cube complexes are important tools in low dimensional topology and group theory and play a prominent role in Agol's proof of the Virtual Haken Conjecture. Constructing a hierarchy for a NPC cube complex is a powerful method of decomposing its fundamental group essential to the theory of NPC cube complex theory. When a cube complex admits a hierarchy with nice properties, it becomes possible to use the hierarchy structure to make inductive arguments. I will explain what a quasiconvex hierarchy of an NPC cube complex is and briefly discuss some of the applications. We will see an outline of how to construct a quasiconvex hierarchy for a relatively hyperbolic NPC cube complex and some of the hyperbolic and relatively hyperbolic geometric tools used to ensure the hierarchy is indeed quasiconvex.
"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.
+
===Marissa Loving===
  
=== Ke Zhu===
+
"Least dilatation of pure surface braids"
"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.
+
The n-stranded pure surface braid group of a genus g surface can be described as the subgroup of the pure mapping class group of a surface of genus g with n-punctures which becomes trivial on the closed surface. I am interested in the least dilatation of pseudo-Anosov pure surface braids. For the n=1 case, upper and lower bounds on the least dilatation were proved by Dowdall and Aougab—Taylor, respectively. In this talk, I will describe the upper and lower bounds I have proved as a function of g and n.
  
=== Shaosai Huang ===
+
===Sara Maloni===
"\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.
+
"On type-preserving representations of thrice punctured projective plane group"
  
=== Sebastian Baader ===
+
In this talk, after a brief overview on famous topological and dynamical open questions on character varieties, we will consider type-preserving representations of the fundamental group of the three-holed projective plane N into PGL(2, R). First, we prove Kashaev’s conjecture on the number of connected components with non-maximal euler class. Second, we show that for all representations with euler class 0 there is a one simple closed curve which is sent to a non-hyperbolic element, while in euler class 1 or -1 we show that there are six components where all the simple closed curves are sent to hyperbolic elements and 2 components where there are some simple closed curves sent to non-hyperbolic elements. This answers a generalisation of a question asked by Bowditch for orientable surfaces. In addition, we show, in most cases, that the action of the pure mapping class group Mod(N) on these non-maximal components is ergodic, proving Goldman conjecture in those cases. Time permitting we will discuss a work in progress with Palesi where we expend these results to all five surfaces (orientable and non-orientable) of characteristic -2. (This is joint work with F. Palesi and T. Yang.)
"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.
+
===Dingxin Zhang===
 +
"Relative cohomology and A-hypergeometric equations"
  
=== Shengwen Wang ===
+
The GKZ hypergeometric equations are closely related to the period integrals of algebraic varieties. Based on the theorems of Walther--Schulze, we identify the set of solutions of a certain GKZ system with some relative homology groups. Our result generalizes the theorem of Huang--Lian--Yau--Zhu. This is a joint work with Tsung-Ju Lee.
"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 ===
+
===Zhongshan An===
"TBA"
+
"Ellipticity of the Bartnik Boundary Conditions"
  
=== Ovidiu Munteanu ===
+
The Bartnik quasi-local mass is defined to measure the mass of a bounded manifold with boundary, where a collection of geometric boundary data — the so-called Bartnik boundary data— plays a key role. Bartnik proposed the open problem whether, on a given manifold with boundary, there exists a stationary vacuum metric so that the Bartnik boundary conditions are realized. In the effort to answer this question, it is important to prove the ellipticity of Bartnik boundary conditions for stationary vacuum metrics. In this talk, I will start with an introduction to the Bartnik quasi-local mass and the moduli space of stationary vacuum metrics. Then I will explain the ellipticity result for the Bartnik boundary conditions and, as an application, give a partial answer to the existence question.
"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.
+
===Xiangdong Xie===
 +
"Quasi-isometric rigidity of a class of right angled Coxeter groups"
  
=== Brian Hepler ===
+
Given any finite simplicial graph G with vertex set V  and edge set E, the associated right angled Coxeter group  (RACG)  W(G) is defined
"Perverse Results on Parameterized Hypersurfaces"
+
as the group with generating set V whose generators all have order 2 and where uv=vu for each edge (u,v).
 +
The classical examples are the reflection groups generated by the reflections about edges of  right angled polygons (in the Euclidean plane or the hyperbolic plane). We classify a class of RACGs up to quasi-isometry.  This is joint work with Jordan Bounds.
  
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.
+
== Spring Abstracts ==
  
 
== Archive of past Geometry seminars ==
 
== Archive of past Geometry seminars ==
 +
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 13:34, 17 April 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


Spring 2019

date speaker title host(s)
April 5 Mark Pengitore (Ohio) Translation-like actions on nilpotent groups (Dymarz)
April 18 José Ignacio Cogolludo Agustín (Universidad de Zaragoza) Even Artin Groups, cohomological computations and other geometrical properties. (Maxim) Unusual date and time: B309 Van Vleck, 2:15-3:15
April 19 Yan Xu (University of Missouri - St. Louis) Structure of minimal two-spheres of constant curvature in hyperquadrics (Huang)

Fall 2018

date speaker title host(s)
Sept. 14 Teddy Einstein (UIC) Quasiconvex Hierarchies for Relatively Hyperbolic Non-Positively Curved Cube Complexes (Dymarz)
Oct. 12 Marissa Loving Least dilatation of pure surface braids (Kent)
Oct. 19 Sara Maloni On type-preserving representations of thrice punctured projective plane group (Kent)
Oct. 26 Dingxin Zhang (Harvard-CMSA) Relative cohomology and A-hypergeometric equations (Huang)
Nov. 9 Zhongshan An (Stony Brook) Ellipticity of the Bartnik Boundary Conditions (Huang)
Nov. 16 Xiangdong Xie Quasi-isometric rigidity of a class of right angled Coxeter groups (Dymarz)


Spring Abstracts

Mark Pengitore

"Translation-like actions on nilpotent groups"

Translation-like actions were introduced Whyte to generalize subgroup containment. Using this notion, he proved that a group is non-amenable if and only if it admits a translation-like action by a non-abelian free group. This result motivates us to ask what groups admit translation-like actions on various interesting classes of groups. As a consequence of Gromov's polynomial growth theorem, we have that only nilpotent groups may act translation-like on a nilpotent group which is the main focus of this talk. Thus, one may ask to characterize what nilpotent groups act translation-like on a fixed nilpotent group. We offer partial answer to this question by demonstrating that if two nilpotent groups have the same growth but distinct asymptotic cones, then there exist no translation-like action of these two groups on each other.

José Ignacio Cogolludo Agustín

"Even Artin Groups, cohomological computations and other geometrical properties."

The purpose of this talk is to introduce even Artin groups and consider their quasi-projectivity properties, as well as study the cohomological properties of their kernels, that is, the kernels of their characters.

Yan Xu

"Structure of minimal two-spheres of constant curvature in hyperquadrics"

Veronese two-sphere (also called rational normal curve) is an interesting projective variety in geometry. It is of constant curvature and unique up to action of unitary group. Based on this rigidity result and SVD (singular value decomposition) in linear algebra, we give a classification of a special class minimal, especially holomorphic, two-spheres of constant curvature in hyperquadric, up to action of real orthogonal group and reparameterization of the two-sphere. For degree less than or equal to three, we give an algorithm and explicit examples. As an application of this results, by computing the norm squared of second fundamental form, we show the generic two-spheres constructed here are not homogeneous. This is a joint work with Professor Quo-Shin Chi and Zhenxiao Xie.

Fall Abstracts

Teddy Einstein

"Quasiconvex Hierarchies for Relatively Hyperbolic Non-Positively Curved Cube Complexes"

Non-positively curved (NPC) cube complexes are important tools in low dimensional topology and group theory and play a prominent role in Agol's proof of the Virtual Haken Conjecture. Constructing a hierarchy for a NPC cube complex is a powerful method of decomposing its fundamental group essential to the theory of NPC cube complex theory. When a cube complex admits a hierarchy with nice properties, it becomes possible to use the hierarchy structure to make inductive arguments. I will explain what a quasiconvex hierarchy of an NPC cube complex is and briefly discuss some of the applications. We will see an outline of how to construct a quasiconvex hierarchy for a relatively hyperbolic NPC cube complex and some of the hyperbolic and relatively hyperbolic geometric tools used to ensure the hierarchy is indeed quasiconvex.

Marissa Loving

"Least dilatation of pure surface braids"

The n-stranded pure surface braid group of a genus g surface can be described as the subgroup of the pure mapping class group of a surface of genus g with n-punctures which becomes trivial on the closed surface. I am interested in the least dilatation of pseudo-Anosov pure surface braids. For the n=1 case, upper and lower bounds on the least dilatation were proved by Dowdall and Aougab—Taylor, respectively. In this talk, I will describe the upper and lower bounds I have proved as a function of g and n.

Sara Maloni

"On type-preserving representations of thrice punctured projective plane group"

In this talk, after a brief overview on famous topological and dynamical open questions on character varieties, we will consider type-preserving representations of the fundamental group of the three-holed projective plane N into PGL(2, R). First, we prove Kashaev’s conjecture on the number of connected components with non-maximal euler class. Second, we show that for all representations with euler class 0 there is a one simple closed curve which is sent to a non-hyperbolic element, while in euler class 1 or -1 we show that there are six components where all the simple closed curves are sent to hyperbolic elements and 2 components where there are some simple closed curves sent to non-hyperbolic elements. This answers a generalisation of a question asked by Bowditch for orientable surfaces. In addition, we show, in most cases, that the action of the pure mapping class group Mod(N) on these non-maximal components is ergodic, proving Goldman conjecture in those cases. Time permitting we will discuss a work in progress with Palesi where we expend these results to all five surfaces (orientable and non-orientable) of characteristic -2. (This is joint work with F. Palesi and T. Yang.)

Dingxin Zhang

"Relative cohomology and A-hypergeometric equations"

The GKZ hypergeometric equations are closely related to the period integrals of algebraic varieties. Based on the theorems of Walther--Schulze, we identify the set of solutions of a certain GKZ system with some relative homology groups. Our result generalizes the theorem of Huang--Lian--Yau--Zhu. This is a joint work with Tsung-Ju Lee.


Zhongshan An

"Ellipticity of the Bartnik Boundary Conditions"

The Bartnik quasi-local mass is defined to measure the mass of a bounded manifold with boundary, where a collection of geometric boundary data — the so-called Bartnik boundary data— plays a key role. Bartnik proposed the open problem whether, on a given manifold with boundary, there exists a stationary vacuum metric so that the Bartnik boundary conditions are realized. In the effort to answer this question, it is important to prove the ellipticity of Bartnik boundary conditions for stationary vacuum metrics. In this talk, I will start with an introduction to the Bartnik quasi-local mass and the moduli space of stationary vacuum metrics. Then I will explain the ellipticity result for the Bartnik boundary conditions and, as an application, give a partial answer to the existence question.

Xiangdong Xie

"Quasi-isometric rigidity of a class of right angled Coxeter groups"

Given any finite simplicial graph G with vertex set V and edge set E, the associated right angled Coxeter group (RACG) W(G) is defined as the group with generating set V whose generators all have order 2 and where uv=vu for each edge (u,v). The classical examples are the reflection groups generated by the reflections about edges of right angled polygons (in the Euclidean plane or the hyperbolic plane). We classify a class of RACGs up to quasi-isometry. This is joint work with Jordan Bounds.

Spring Abstracts

Archive of past Geometry seminars

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