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 ==
== Spring 2017 ==
 
 
 
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!align="left" | date
 
!align="left" | date
Line 15: Line 13:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|Jan 20
+
|April  5
| [http://people.mpim-bonn.mpg.de/rovi/ Carmen Rovi] (University of Indiana Bloomington)
+
|Mark Pengitore (Ohio)
| [[#Carmen Rovi| "The mod 8 signature of a fiber bundle"]]
+
|Translation-like actions on nilpotent groups
| Maxim
+
 
 +
|(Dymarz)
 
|-
 
|-
|Jan 27
+
|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'''
 
|-
 
|-
|Feb 3
+
 
| Rafael Montezuma (University of Chicago)  
+
|April 19
| [[#Rafael Montezuma| "Metrics of positive scalar curvature and unbounded min-max widths"]]
+
|Yan Xu (University of Missouri - St. Louis)
| Lu Wang
+
|Structure of minimal two-spheres of constant curvature in hyperquadrics
|-
+
|(Huang)
|Feb 10
+
 
+
|}
|
+
 
|  
+
== Fall 2018 ==
|-
+
 
|Feb 17
+
{| cellpadding="8"
|[http://www.math.northwestern.edu/~hartman/ Yair Hartman] (Northwestern University) 
+
!align="left" | date
|[[#Yair Hartman| "Intersectional Invariant Random Subgroups and Furstenberg Entropy."]]
+
!align="left" | speaker
| [http://www.math.wisc.edu/~dymarz Dymarz]
+
!align="left" | title
 +
!align="left" | host(s)
 
|-
 
|-
|Feb 24
+
|Sept. 14
| [http://math.uchicago.edu/~lambrozio/ Lucas Ambrozio] (University of Chicago)
+
|Teddy Einstein (UIC)
| [[#Lucas Ambrozio| "Free boundary minimal hypersurfaces of Euclidean domains"]]
+
|Quasiconvex Hierarchies for Relatively Hyperbolic Non-Positively Curved Cube Complexes
| Lu Wang
+
|(Dymarz)
 
|-
 
|-
|March 3
+
|Oct. 12
| [http://www.math.uqam.ca/~powell/ Mark Powell] (Université du Québec à Montréal)
+
|Marissa Loving
| [[#Mark Powell| "Stable classification of 4-manifolds"]]
+
|Least dilatation of pure surface braids
| Kjuchukova
+
|(Kent)
 
|-
 
|-
|March 10
+
|Oct. 19
| [http://www.math.wisc.edu/~kent Autumn Kent] (Wisconsin)
+
|Sara Maloni
| [[#Autumn Kent | ''Analytic functions from hyperbolic manifolds'']]
+
|On type-preserving representations of thrice punctured projective plane group
| local
+
|(Kent)
 
|-
 
|-
|March 17
+
|Oct. 26
|  
+
|Dingxin Zhang (Harvard-CMSA)
|  
+
|Relative cohomology and A-hypergeometric equations
|  
+
|(Huang)
 
|-
 
|-
|March 24
+
|Nov. 9
| Spring Break
+
|Zhongshan An (Stony Brook)
|  
+
|Ellipticity of the Bartnik Boundary Conditions
|  
+
|(Huang)
 
|-
 
|-
|March 31
+
|Nov. 16
| [http://www.math.uci.edu/~xiangwen/ Xiangwen Zhang] (University of California-Irvine)
+
|Xiangdong Xie
| [[#Xiangwen Zhang| "The Anomaly Flow and Strominger systems"]]
+
|Quasi-isometric rigidity of a class of right angled Coxeter groups
| Lu Wang
+
|(Dymarz)
|  
 
|-
 
|April 7
 
| [https://www.math.rutgers.edu/~feehan/ Paul Feehan] (Rutgers)
 
|[[#Paul Feehan| "TBA"]]
 
| Lu Wang
 
|-
 
|April 14
 
| [https://www.math.wisc.edu/~gong/ Xianghong Gong] (Wisconsin)
 
| [[#Xianghong Gong| "TBA"]]
 
| local
 
|-
 
|April 21
 
|[http://www.math.csi.cuny.edu/~maher/ Joseph Maher] (CUNY) 
 
| [[#Joseph Maher|"TBA"]]
 
| [http://www.math.wisc.edu/~dymarz Dymarz]
 
|-
 
|April 28
 
| [http://bena-tshishiku.squarespace.com/ Bena Tshishiku] (Harvard)
 
| [[#Bena Tshishiku| "TBA"]]
 
| [http://www.math.wisc.edu/~dymarz Dymarz]
 
 
|-
 
|-
 
|
 
|
 
|}
 
|}
  
== Spring Abstracts ==
 
  
===Lucas Ambrozio===
+
==Spring Abstracts==
"Free boundary minimal hypersurfaces of Euclidean domains"
+
 
 +
===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"
  
We will show how the first Betti number of a compact free boundary minimal hypersurface in an domain whose boundary satisfies weak convexity assumptions is controlled effectively by the Morse index of this hypersurface viewed as a critical point of the area functional (joint with A. Carlotto and B. Sharp). Among such domains, the unit three-ball is particularly interesting, as it contains many free boundary minimal surfaces, which one would like to classify. In particular, we will explain how to characterise the critical catenoid in terms of a pinching condition on the second fundamental form (joint with I. Nunes).
+
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.
  
===Paul Feehan===
+
===Marissa Loving===
"TBA"
 
  
===Rafael Montezuma===
+
"Least dilatation of pure surface braids"
"Metrics of positive scalar curvature and unbounded min-max widths"
 
  
In this talk, I will construct a sequence of Riemannian metrics on the three-dimensional sphere with scalar curvature greater than or equal to 6, and arbitrarily large min-max widths. The search for such metrics is motivated by a rigidity result of min-max minimal spheres in three-manifolds obtained by Marques and Neves.
+
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.
  
===Carmen Rovi===
+
===Sara Maloni===
''The mod 8 signature of a fiber bundle''
 
  
In this talk we shall be concerned with the residues modulo 4 and modulo 8 of the signature of a 4k-dimensional geometric Poincare complex. I will explain the relation between the signature modulo 8 and two other invariants: the Brown-Kervaire invariant and the Arf invariant. In my thesis I applied the relation between these invariants to the study of the signature modulo 8 of a fiber bundle. In 1973 Werner Meyer used group cohomology to show that a surface bundle has signature divisible by 4. I will discuss current work with David Benson, Caterina Campagnolo and Andrew Ranicki where we are using group cohomology and  representation theory of finite groups to detect non-trivial signatures modulo 8 of surface bundles.
+
"On type-preserving representations of thrice punctured projective plane group"
  
===Yair Hartman===
+
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.)
"Intersectional Invariant Random Subgroups and Furstenberg Entropy."
 
  
In this talk I'll present a joint work with Ariel Yadin, in which we solve the Furstenberg Entropy Realization Problem for finitely supported random walks (finite range jumps) on free groups and lamplighter groups. This generalizes a previous result of Bowen. The proof consists of several reductions which have geometric and probabilistic flavors of independent interests.
+
===Dingxin Zhang===
All notions will be explained in the talk, no prior knowledge of Invariant Random Subgroups or Furstenberg Entropy is assumed.
+
"Relative cohomology and A-hypergeometric equations"
  
===Bena Tshishiku===
+
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.
"TBA"
 
  
===Mark Powell===
 
''Stable classification of 4-manifolds''
 
  
A stabilisation of a 4-manifold M is a connected sum of M with some number of copies of S^2 x S^2.
+
===Zhongshan An===
Two 4-manifolds are said to be stably diffeomorphic if they admit diffeomorphic stabilisations.
+
"Ellipticity of the Bartnik Boundary Conditions"
Since a necessary condition is that the fundamental groups be isomorphic, we study this equivalence relation for a fixed group.  I will discuss recent progress in classifying 4-manifolds up to stable diffeomorphism for certain families of groups, arising from work with Daniel Kasprowski, Markus Land and Peter Teichner. 
 
As a by-product we also obtained a result on the analogous question with the complex projective plane CP^2 replacing S^2 x S^2.
 
  
===Autumn Kent===
+
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.
''Analytic functions from hyperbolic manifolds''
 
  
At the heart of Thurston's proof of Geometrization for Haken manifolds is a family of analytic functions between Teichmuller spaces called "skinning maps."  These maps carry geometric information about their associated hyperbolic manifolds, and I'll discuss what is presently known about their behavior.  The ideas involved form a mix of geometry, algebra, and analysis.
+
===Xiangdong Xie===
 +
"Quasi-isometric rigidity of a class of right angled Coxeter groups"
  
===Xiangwen Zhang===
+
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
"The Anomaly Flow and Strominger systems"
+
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.
  
The anomaly flow is a geometric flow which implements the Green-Schwarz anomaly cancellation mechanism originating from superstring theory, while preserving the conformally balanced condition of Hermitian metrics. I will discuss criteria for long time existence and convergence of the flow on toric fibrations with the Fu-Yau ansatz. This is joint work with D.H. Phong and S. Picard.
+
== 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