Geometry and Topology Seminar 2019-2020: Difference between revisions

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The [[Geometry and Topology]] seminar meets in room '''B223 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]]


== Fall 2016 ==
 
== Fall 2018 ==


{| cellpadding="8"
{| cellpadding="8"
Line 13: Line 14:
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|September 9
|Sept. 14
| [http://www.math.wisc.edu/~bwang/ Bing Wang] (UW Madison)
|Teddy Einstein (UIC)
| [[#Bing Wang| "The extension problem of the mean curvature flow"]]
|Quasiconvex Hierarchies for Relatively Hyperbolic Non-Positively Curved Cube Complexes
| (Local)
|(Dymarz)
|-
|-
|September 16
|Oct. 12
| [http://www.math.northwestern.edu/~weinkove/ Ben Weinkove] (Northwestern University)
|Marissa Loving
| [[#Ben Weinkove| Gauduchon metrics with prescribed volume form]]
|Least dilatation of pure surface braids
| Lu Wang
|(Kent)
|-
|-
|September 23
|Oct. 19
| Jiyuan Han (UW Madison)
|Sara Maloni
| [[#Jiyuan Han| "TBA"]]
|On type-preserving representations of thrice punctured projective plane group
| (Local)
|(Kent)
|-
|-
|September 30
|Oct. 26
|  
|Dingxin Zhang (Harvard-CMSA)
|  
|Relative cohomology and A-hypergeometric equations
|
|(Huang)
|-
|-
|October 7
|Nov. 9
| Yu Li (UW Madison)
|Zhongshan An (Stony Brook)
| [[#Yu Li| "TBA"]]
|Ellipticity of the Bartnik Boundary Conditions
| (Local)
|(Huang)
|-
|-
|October 14
|Nov. 16
| [http://math.uchicago.edu/~seanpkh/ Sean Howe] (University of Chicago)
|Xiangdong Xie
| [[#Sean Howe| "TBA"]]
|TBA
| Melanie Matchett Wood
|(Dymarz)
|-
|October 21
|
|
|-
|October 28
| Ronan Conlon
| [[#Ronan Conlon| "TBA"]]
| Bing Wang
|-
|November 4
| Jonathan Zhu (Harvard University)
| [[#Jonathan Zhu| "TBA"]]
| Lu Wang
|-
|'''November 7'''
| [http://www.massey.ac.nz/massey/expertise/profile.cfm?stref=339830 Gaven Martin] (University of New Zealand)
| [[#Gaven Martin| "TBA"]]
| Simon Marshall
|-
|November 11
|
|
|-
|November 18
|
|
|-
| Thanksgiving Recess
|
|
|
|-
|December 2
|Peyman Morteza (UW Madison)
| [[#Peyman Morteza| "TBA"]]
| (Local)
|-
|December 9
|
|
|-
|December 16
|
|
|-
|-
|
|
Line 98: Line 49:
== Fall Abstracts ==
== Fall Abstracts ==


=== Ronan Conlon ===
===Teddy Einstein===
''TBA''
 
"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.


=== Jiyuan Han ===
===Sara Maloni===
''TBA''


=== Sean Howe ===
"On type-preserving representations of thrice punctured projective plane group"
''TBA''


===Yu Li ===
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.)
''TBA''


===Gaven Marin ===
===Dingxin Zhang===
''TBA''
"Relative cohomology and A-hypergeometric equations"


===Peyman Morteza ===
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''


=== Bing Wang ===
''The extension problem of the mean curvature flow''


===Zhongshan An===
"Ellipticity of the Bartnik Boundary Conditions"


We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R^3.
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.  
A key ingredient of the proof is to show a two-sided pseudo-locality property of the mean curvature flow, whenever the mean curvature is bounded.
This is a joint work with Haozhao Li.


=== Ben Weinkove ===
Every compact complex manifold admits a Gauduchon metric in each conformal class of Hermitian metrics. In 1984 Gauduchon conjectured that one can prescribe the volume form of such a metric.  I will discuss the proof of this conjecture, which amounts to solving a nonlinear Monge-Ampere type equation.  This is a joint work with Gabor Szekelyhidi and Valentino Tosatti.


=== Jonathan Zhu ===
''TBA''


== 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]]
<br><br>
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]
<br><br>
<br><br>

Revision as of 04:06, 14 October 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 Shaosai Huang.

Hawk.jpg


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 TBA (Dymarz)

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.


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