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/~dymarz Tullia Dymarz] or [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova].
+
For more information, contact Shaosai Huang.
  
 
[[Image:Hawk.jpg|thumb|300px]]
 
[[Image:Hawk.jpg|thumb|300px]]
 
== Summer 2015 ==
 
  
  
 +
== Spring 2019 ==
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!align="left" | date
 
!align="left" | date
Line 14: Line 13:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|<b>June 23 at 2pm in Van Vleck 901</b>
+
|April  5
| [http://www2.warwick.ac.uk/fac/sci/maths/people/staff/david_epstein/ David Epstein] (Warwick)
+
|Mark Pengitore (Ohio)
| [[#David Epstein (Warwick) |''Splines and manifolds.'']]
+
|Translation-like actions on nilpotent groups
| Hirsch
+
 
 +
|(Dymarz)
 
|-
 
|-
|}
+
|April  18
 +
|José Ignacio Cogolludo Agustín (Universidad de Zaragoza)
 +
|Even Artin Groups, cohomological computations and other geometrical properties.
  
== Summer Abstracts ==
+
|(Maxim)
 +
|'''Unusual date and time: B309 Van Vleck, 2:15-3:15'''
 +
|-
  
===David Epstein (Warwick)===
+
|April 19
''Splines and manifolds.''
+
|Yan Xu (University of Missouri - St. Louis)
 
+
|Structure of minimal two-spheres of constant curvature in hyperquadrics
[http://www.math.wisc.edu/~rkent/Abstract.Epstein.2015.pdf Abstract (pdf)]
+
|(Huang)
 
 
 
 
 
 
== Fall 2015==
 
  
 +
|}
  
 +
== Fall 2018 ==
  
 
{| cellpadding="8"
 
{| cellpadding="8"
Line 40: Line 42:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|September 4
+
|Sept. 14
|
+
|Teddy Einstein (UIC)
|
+
|Quasiconvex Hierarchies for Relatively Hyperbolic Non-Positively Curved Cube Complexes
|
+
|(Dymarz)
|-
 
|September 11
 
| [https://uwm.edu/math/people/tran-hung-1/ Hung Tran] (UW Milwaukee)
 
| [[#Hung Tran|''Relative divergence, subgroup distortion, and geodesic divergence'']]
 
| [http://www.math.wisc.edu/~dymarz T. Dymarz]
 
|-
 
|September 18
 
| [http://www.math.wisc.edu/~dymarz Tullia Dymarz] (UW Madison)
 
| [[#Tullia Dymarz|''Non-rectifiable Delone sets in amenable groups'']]
 
| (local)
 
|-
 
|September 25
 
| [https://jpwolfson.wordpress.com/ Jesse Wolfson] (Uchicago)
 
| [[#Jesse Wolfson|''Counting Problems and Homological Stability'']]
 
| [http://www.math.wisc.edu/~mmwood/ M. Matchett Wood]
 
|-
 
|October 2
 
| [https://riemann.unizar.es/~jicogo/ Jose Ignacio Cogolludo Agustín] (University of Zaragoza, Spain)
 
| [[#Jose Ignacio Cogolludo Agustín|''Topology of curve complements and combinatorial aspects'']]
 
|[http://www.math.wisc.edu/~maxim L. Maxim]
 
 
|-
 
|-
|October 9
+
|Oct. 12
| [http://people.brandeis.edu/~mcordes/ Matthew Cordes] (Brandeis)
+
|Marissa Loving
| [[#Matthew Cordes|''Morse boundaries of geodesic metric spaces'']]
+
|Least dilatation of pure surface braids
| [http://www.math.wisc.edu/~dymarz T. Dymarz]
+
|(Kent)
 
|-
 
|-
|October 16
+
|Oct. 19
| [http://www.math.jhu.edu/~bernstein/ Jacob Bernstein] (Johns Hopkins University)
+
|Sara Maloni
| [[#Jacob Bernstein (Johns Hopkins University)|''Hypersurfaces of low entropy'']]
+
|On type-preserving representations of thrice punctured projective plane group
| [http://www.sites.google.com/a/wisc.edu/lu-wang/ L. Wang]
+
|(Kent)
 
|-
 
|-
|October 23
+
|Oct. 26
| [https://sites.google.com/a/wisc.edu/ysu/ Yun Su] (UW Madison)
+
|Dingxin Zhang (Harvard-CMSA)
| [[#Yun Su (Brandeis)|''Higher-order degrees of hypersurface complements.'']]
+
|Relative cohomology and A-hypergeometric equations
| (local)
+
|(Huang)
 
|-
 
|-
|October 30
+
|Nov. 9
| [http://www.math.stonybrook.edu/phd-student-directory Gao Chen] (Stony Brook University)
+
|Zhongshan An (Stony Brook)
| [[#Gao Chen(Stony Brook University)|''Classification of gravitational instantons '']]
+
|Ellipticity of the Bartnik Boundary Conditions
| [http://www.math.wisc.edu/~bwang B.Wang]
+
|(Huang)
 
|-
 
|-
|November 6
+
|Nov. 16
| [http://scholar.harvard.edu/gardiner Dan Cristofaro-Gardiner] (Harvard)
+
|Xiangdong Xie
| [[#Dan Cristofaro-Gardiner|''Higher-dimensional symplectic embeddings and the Fibonacci staircase'']]
+
|Quasi-isometric rigidity of a class of right angled Coxeter groups
| [http://www.math.wisc.edu/~kjuchukova Kjuchukova]
+
|(Dymarz)
|
 
|-
 
|November 13
 
| [http://people.brandeis.edu/~ruberman/ Danny Ruberman] (Brandeis)
 
| [[#Danny Ruberman|''Configurations of embedded spheres'']]
 
| [http://www.math.wisc.edu/~kjuchukova Kjuchukova]
 
|
 
|-
 
|November 20
 
| [https://www.math.toronto.edu/cms/izosimov-anton/ Anton Izosimov] (University of Toronto)
 
| [[#Anton Izosimov (University of Toronto)|''TBA'']]
 
| [http://www.math.wisc.edu/~maribeff/ Mari-Beffa]
 
|-
 
|Thanksgiving Recess
 
|
 
|
 
|
 
|-
 
|December 4
 
| [http://www.math.wisc.edu/~westrich/ Quinton Westrich] (UW Madison)
 
| [[#Quinton Westrich (UW Madison) |''Harmonic Chern Forms on Polarized Kähler Manifolds'']]
 
| (local)
 
|-
 
|December 11
 
|[http://kaihowong.weebly.com/ Tommy Wong] (UW Madison)
 
| [[#Tommy Wong (UW Madison)|''Milnor Fiber of Complex Hyperplane Arrangement.'']]
 
| (local)
 
 
|-
 
|-
 
|
 
|
 
|}
 
|}
  
== Fall Abstracts ==
 
  
 +
==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.
  
===Hung Tran===
+
===Yan Xu===
''Relative divergence, subgroup distortion, and geodesic divergence''
+
"Structure of minimal two-spheres of constant curvature in hyperquadrics"
  
In my presentation, I introduce three new invariants for pairs $(G;H)$ consisting of a finitely generated group $G$ and a subgroup $H$. The first invariant is the upper relative divergence which generalizes Gersten's notion of divergence. The second invariant is the lower relative divergence which generalizes a definition of Cooper-Mihalik. The third invariant is the lower subgroup distortion which parallels the standard notion
+
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.
of subgroup distortion. We examine the relative divergence (both upper and lower) of a group with respect to a normal subgroup or a cyclic subgroup. We also explore relative divergence of $CAT(0)$ groups and relatively hyperbolic groups with respect to various subgroups to better understand geometric properties of these groups. We answer the question of Behrstock and Drutu about the existence of Morse geodesics in $CAT(0)$ spaces with divergence function strictly greater than $r^n$ and strictly less than $r^{n+1}$, where $n$ is an integer greater than $1$. More precisely, we show that for each real number $s>2$, there is a $CAT(0)$ space $X$ with a proper and cocompact action of some finitely generated group such that $X$ contains a Morse bi-infinite geodesic with the divergence equivalent to $r^s$.
 
  
 +
== Fall Abstracts ==
  
===Tullia Dymarz===
+
===Teddy Einstein===
''Non-rectifiable Delone sets in amenable groups''
 
  
In 1998 Burago-Kleiner and McMullen constructed the first
+
"Quasiconvex Hierarchies for Relatively Hyperbolic Non-Positively Curved Cube Complexes"
examples of coarsely dense and uniformly discrete subsets of R^n that are
 
not biLipschitz equivalent to the standard lattice Z^n. Similarly we
 
find subsets inside the three dimensional solvable Lie group SOL that are
 
not bilipschitz to any lattice in SOL. The techniques involve combining
 
ideas from Burago-Kleiner with quasi-isometric rigidity results from
 
geometric group theory.
 
  
===Jesse Wolfson===
+
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.
''Counting Problems and Homological Stability''
 
  
In 1969, Arnold showed that the i^{th} homology of the space of un-ordered configurations of n points in the plane becomes independent of n for n>>i. A decade later, Segal extended Arnold's method to show that the i^{th} homology of the space of degree n holomorphic maps from \mathbb{P}^1 to itself also becomes independent of n for large n, and, moreover, that both sequences of spaces have the same limiting homology. We explain how, using Weil's number field/function field dictionary, one might have predicted this topological coincidence from easily verifiable statements about specific counting problems.  We then discuss ongoing joint work with Benson Farb and Melanie Wood in which we use other counting problems to predict and discover new instances of homological stability in the topology of complex manifolds.
+
===Marissa Loving===
  
 +
"Least dilatation of pure surface braids"
  
===Matthew Cordes===
+
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.
''Morse boundaries of geodesic metric spaces''
 
  
I will introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with equivalence classes of geodesic rays that identify the ``hyperbolic directions" in that space. (A ray is Morse if quasi-geodesics with endpoints on the ray stay bounded distance from the ray.) This boundary is a quasi-isometry invariant and a visibility space. In the case of a proper CAT(0) space the Morse boundary generalizes the contracting boundary of Charney and Sultan and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. Time permitting I will also discuss some results on Morse boundary of the mapping class group and briefly describe joint work with David Hume developing a capacity dimension for the Morse boundary.
+
===Sara Maloni===
  
===Anton Izosimov===
+
"On type-preserving representations of thrice punctured projective plane group"
''TBA''
 
  
===Jacob Bernstein===
+
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.)
''Hypersurfaces of low entropy''
 
  
The entropy is a quantity introduced by Colding and Minicozzi and may be thought of as a rough measure of the geometric complexity of a hypersurface of Euclidean space.  It is closely related to the mean curvature flow.  On the one hand, the entropy controls the dynamics of the flow. On the other hand, the mean curvature flow may be used to study the entropy.  In this talk I will survey some recent results with Lu Wang that show that hypersurfaces of low entropy really are simple.
+
===Dingxin Zhang===
 +
"Relative cohomology and A-hypergeometric equations"
  
===Yun Su===
+
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.
''Higher-order degrees of hypersurface complements.''
 
  
===Gao Chen===
 
''Classification of gravitational instantons''
 
  
A gravitational instanton is a noncompact complete hyperkahler manifold of real dimension 4 with faster than quadratic curvature decay. In this talk, I will discuss the recent work towards the classification of gravitational instantons. This is a joint work with X. X. Chen.
+
===Zhongshan An===
 +
"Ellipticity of the Bartnik Boundary Conditions"
  
===Dan Cristofaro-Gardiner===
+
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.
''Higher-dimensional symplectic embeddings and the Fibonacci staircase''
 
  
McDuff and Schlenk determined when a four dimensional symplectic ellipsoid can be embedded into a ball, and found that when the ellipsoid is close to round, the answer is given by an infinite staircase determined by the odd-index Fibonacci numbers.  I will explain joint work with Richard Hind, showing that a generalization of this holds in all even dimensions.
+
===Xiangdong Xie===
 +
"Quasi-isometric rigidity of a class of right angled Coxeter groups"
  
===Danny Ruberman===
+
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
''Configurations of embedded spheres''
+
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.
  
Configurations of lines in the plane have been studied since antiquity. In recent years, combinatorial methods have been used to decide if a specified incidence relation between certain objects ("lines") and other objects ("points") can be realized by actual points and lines in a projective plane over a field. For the real and complex fields, one can weaken the condition to look for topologically embedded lines (circles in the real case, spheres in the complex case) that meet according to a specified incidence relation. I will explain some joint work with Laura Starkston (Stanford) giving new topological restrictions on the realization of configurations of spheres in the complex projective plane.
+
== 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]]
 +
<br><br>
 +
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]
 +
<br><br>
 
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]
 
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]
 
<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