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 2018 ==
+
== Fall 2018 ==
  
 
{| cellpadding="8"
 
{| cellpadding="8"
Line 14: Line 14:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|January 26
+
|Sept. 14
|Jingrui Cheng
+
|Teddy Einstein (UIC)
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence"]]
+
|Quasiconvex Hierarchies for Relatively Hyperbolic Non-Positively Curved Cube Complexes
|Local
+
|(Dymarz)
 
|-
 
|-
|February 2
+
|Oct. 12
|Jingrui Cheng
+
|Marissa Loving
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence" (continued)]]
+
|Least dilatation of pure surface braids
|Local
+
|(Kent)
 
|-
 
|-
|February 9
+
|Oct. 19
|Jingrui Cheng
+
|Sara Maloni
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence" (continued)]]
+
|On type-preserving representations of thrice punctured projective plane group
|Local
+
|(Kent)
 
|-
 
|-
|February 16
+
|Oct. 26
|TBA
+
|Dingxin Zhang (Harvard-CMSA)
|TBA
+
|Relative cohomology and A-hypergeometric equations
|TBA
+
|(Huang)
 
|-
 
|-
|February 23
+
|Nov. 9
|TBA
+
|Zhongshan An (Stony Brook)
|TBA
+
|Ellipticity of the Bartnik Boundary Conditions
|TBA
+
|(Huang)
 
|-
 
|-
|March 2
+
|Nov. 16
|TBA
+
|Xiangdong Xie
|TBA
+
|TBA
+
|-
+
|March 9
+
|TBA
+
|TBA
+
|TBA
+
|-
+
|March 16
+
|Yu Li
+
|TBA
+
|Bing Wang
+
|-
+
|March 23
+
|TBA
+
|TBA
+
|TBA
+
|-
+
|<b> Spring Break </b>
+
|
+
|
+
|-
+
|April 6
+
|Wei Ho
+
|TBA
+
|Daniel Erman
+
|-
+
|April 13
+
|TBA
+
|TBA
+
|TBA
+
|-
+
|April 20
+
|TBA
+
|TBA
+
|TBA
+
|-
+
|April 27
+
|TBA
+
|TBA
+
|TBA
+
|-
+
|May 4
+
|TBA
+
|TBA
+
 
|TBA
 
|TBA
 +
|(Dymarz)
 
|-
 
|-
 
|
 
|
|}
 
 
== 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.
 
 
 
== Fall 2017 ==
 
 
{| cellpadding="8"
 
!align="left" | date
 
!align="left" | speaker
 
!align="left" | title
 
!align="left" | host(s)
 
|-
 
|September 8
 
|TBA
 
|TBA
 
|TBA
 
|-
 
|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
 
|Marco Mendez-Guaraco (Chicago)
 
|[[#Marco Mendez-Guaraco| "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)
 
|[[#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| "Deformation Formulas for Parameterizable Hypersurfaces"]]
 
|Max
 
|-
 
 
|}
 
|}
  
 
== 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
+
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.
+
"Quasiconvex Hierarchies for Relatively Hyperbolic Non-Positively Curved Cube Complexes"
  
=== Ke Zhu===
+
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.
"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.
+
===Marissa Loving===
  
=== Shaosai Huang ===
+
"Least dilatation of pure surface braids"
"\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.  
+
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.
  
=== Sebastian Baader ===
+
===Sara Maloni===
"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.
+
"On type-preserving representations of thrice punctured projective plane group"
  
=== Shengwen Wang ===
+
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.)
"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.
+
===Dingxin Zhang===
 +
"Relative cohomology and A-hypergeometric equations"
  
=== Marco Mendez-Guaraco ===
+
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.
"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 ===
+
===Zhongshan An===
"The geometry of four dimensional shrinking Ricci solitons"
+
"Ellipticity of the Bartnik Boundary Conditions"
  
I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons.  
+
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.  
  
=== 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.
 
  
 
== 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 23:06, 13 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