Geometry and Topology Seminar 2019-2020: Difference between revisions

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== Fall 2018 ==
== Spring 2020 ==


{| cellpadding="8"
{| cellpadding="8"
Line 14: Line 14:
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|Sept. 14
|Feb. 7
|Teddy Einstein (UIC)
|Xiangdong Xie  (Bowling Green University)
|Quasiconvex Hierarchies for Relatively Hyperbolic Non-Positively Curved Cube Complexes
| Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|(Dymarz)
|(Dymarz)
|-
|-
|Oct. 12
|Feb. 14
|Marissa Loving
|Xiangdong Xie  (Bowling Green University)
|Least dilatation of pure surface braids
| Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|(Kent)
|(Dymarz)
|-
|-
|Oct. 19
|Feb. 21
|Sara Maloni
|Xiangdong Xie  (Bowling Green University)
|On type-preserving representations of thrice punctured projective plane group
| Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|(Kent)
|(Dymarz)
|-
|-
|Oct. 26
|Feb. 28
|Dingxin Zhang (Harvard-CMSA)
|Kuang-Ru Wu (Purdue University)
|Relative cohomology and A-hypergeometric equations
|Griffiths extremality, interpolation of norms, and Kahler quantization
|(Huang)
|(Huang)
|-
|-
|Nov. 9
|Mar. 6
|Zhongshan An (Stony Brook)
|Yuanqi Wang (University of Kansas)
|Ellipticity of the Bartnik Boundary Conditions
|Moduli space of G2−instantons on 7−dimensional product manifolds
|(Huang)
|(Huang)
|-
|-
|Nov. 16
|Mar. 13 <b>CANCELED</b>
|Xiangdong Xie
|Karin Melnick (University of Maryland)
|Quasi-isometric rigidity of a class of right angled Coxeter groups
|A D'Ambra Theorem in conformal Lorentzian geometry
|(Dymarz)
|-
|<b>Mar. 25</b> <b>CANCELED</b>
|Joerg Schuermann (University of Muenster, Germany)
|An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles
|(Maxim)
|-
|Mar. 27 <b>CANCELED</b>
|David Massey (Northeastern University)
|Extracting easily calculable algebraic data from the vanishing cycle complex
|(Maxim)
|-
|<b>Apr. 10</b> <b>CANCELED</b>
|Antoine Song (Berkeley)
|TBA
|(Chen)
|}
 
== Fall 2019 ==
 
{| cellpadding="8"
!align="left" | date
!align="left" | speaker
!align="left" | title
!align="left" | host(s)
|-
|Oct. 4
|Ruobing Zhang (Stony Brook University)
| Geometric analysis of collapsing Calabi-Yau spaces
|(Chen)
|-
|-
|Oct. 25
|Emily Stark (Utah)
|  Action rigidity for free products of hyperbolic manifold groups
|(Dymarz)
|-
|Nov. 8
|Max Forester (University of Oklahoma)
|Spectral gaps for stable commutator length in some cubulated groups
|(Dymarz)
|(Dymarz)
|-
|-
|
|Nov. 22
|Yu Li (Stony Brook University)
|On the structure of Ricci shrinkers
|(Huang)
|-
|}
|}


== Fall Abstracts ==
==Spring Abstracts==
 
===Xiangdong Xie===


===Teddy Einstein===
The quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces has played
an important role in various  rigidity questions in geometry and group theory.
In these talks I  shall give an introduction to this topic.  In the first talk I will introduce Gromov hyperbolic spaces, define their ideal boundary,  and  discuss their basic properties.  In the second and third talks I will define the visual metrics on the ideal boundary, explain the connection between quasiisometries of  Gromov hyperbolic space and quasiconformal maps on  their ideal boundary, and indicate  how the quasiconformal structure on the ideal boundary can be used to deduce rigidity.


"Quasiconvex Hierarchies for Relatively Hyperbolic Non-Positively Curved Cube Complexes"
===Kuang-Ru Wu===


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.
Following Kobayashi, we consider Griffiths negative complex Finsler bundles, naturally leading us to introduce Griffiths extremal Finsler metrics. As we point out, this notion is closely related to the theory of interpolation of norms, and is characterized by an equation of complex Monge– Ampere type, whose corresponding Dirichlet problem we solve. As applications, we prove that Griffiths extremal Finsler metrics quantize solutions to a natural PDE in Kahler geometry, related to the construction of flat maps for the Mabuchi metric. This is joint work with Tamas Darvas.


===Marissa Loving===
===Yuanqi Wang===
$G_{2}-$instantons are 7-dimensional analogues of flat connections in dimension 3. It is part of Donaldson-Thomas’ program to generalize the fruitful gauge theory in dimensions 2,3,4 to dimensions 6,7,8. The moduli space of  $G_{2}-$instantons, with virtual dimension $0$, is  expected to have interesting  geometric structure and  yield enumerative invariant for the underlying $7-$dimensional manifold.


"Least dilatation of pure surface braids"
In this talk, in some reasonable special cases and a fairly complete manner, we will describe the relation between the moduli space of $G_{2}-$instantons and an algebraic geometry moduli on a Calabi-Yau 3-fold.


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.
===Karin Melnick===


===Sara Maloni===
D'Ambra proved in 1988 that the isometry group of a compact, simply connected, real-analytic Lorentzian manifold must be compact. I will discuss my recent theorem that the conformal group of such a manifold must also be compact, and how it relates to the Lorentzian Lichnerowicz Conjecture.


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


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.)
We give an introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles, based on stratified Morse theory for constructible functions. The corresponding local index of an isolated critical point (in a stratified sense) of a one-form depends on the constructible function, specializing for different choices to well known indices like the radial, GSV or Euler obstruction index.


===Dingxin Zhang===
===David Massey===
"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.
Given a complex analytic function on an open subset U  of C<sup>n+1</sup>, one may consider the complex of sheaves of vanishing cycles along f of the constant sheaf Z<sub>U</sub>. This complex encodes on the cohomological level the reduced cohomology of the Milnor fibers of f at each of f<sup>-1</sup>(0). The question is: how does one calculate (ideally, by hand)  any useful numbers about this vanishing cycle complex? One answer is to look at the Lê numbers of f. We will discuss the precise relationship between these objects/numbers.


===Antoine Song===


===Zhongshan An===
TBA
"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.
==Fall Abstracts==


===Xiangdong Xie===
===Ruobing Zhang===
"Quasi-isometric rigidity of a class of right angled Coxeter groups"
 
This talk centers on the degenerations of Calabi-Yau metrics. We will focus on the interactions between algebraic degenerations and metric convergence with highly singular behaviors in the collapsing case. As the complex structures degenerate, the collapsing Calabi-Yau metrics may exhibit various wild geometric properties with highly non-algebraic features.
 
First, as motivating examples, we will describe our recent results on the new collapsing mechanisms of K3 surfaces. Next, we will switch to higher dimensions and we will exhibit some entirely new constructions of degenerating Calabi-Yau metrics which are expected to work in broader contexts. Complex structures degeneration will be accurately characterized by the bubbling and singularity analysis in a geometric manner.
 
===Emily Stark===
 
The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold group is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.
 
===Max Forester===
 
I will discuss stable commutator length (scl) in groups, and some gap theorems for the scl spectrum. Such results say that for various groups, scl of an element is always either zero or is larger than some uniform constant. I will discuss the cases of right-angled Artin groups and certain right-angled Coxeter groups. This is joint work with Pallavi Dani, Ignat Soroko, and Jing Tao.


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 by
===Yu Li===
as the group with generating set V whose generators all have order 2 and where uv=vu for each edge (u,v).
We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As an application, we show that any Ricci shrinker whose second eigenvalue of the curvature operator is positive must be a quotient of sphere.
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.


== Archive of past Geometry seminars ==
== Archive of past Geometry seminars ==
2018-2019  [[Geometry_and_Topology_Seminar_2018-2019]]
<br><br>
2017-2018 [[Geometry_and_Topology_Seminar_2017-2018]]
2017-2018 [[Geometry_and_Topology_Seminar_2017-2018]]
<br><br>
<br><br>

Latest revision as of 18:56, 3 September 2020

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 2020

date speaker title host(s)
Feb. 7 Xiangdong Xie (Bowling Green University) Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces (Dymarz)
Feb. 14 Xiangdong Xie (Bowling Green University) Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces (Dymarz)
Feb. 21 Xiangdong Xie (Bowling Green University) Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces (Dymarz)
Feb. 28 Kuang-Ru Wu (Purdue University) Griffiths extremality, interpolation of norms, and Kahler quantization (Huang)
Mar. 6 Yuanqi Wang (University of Kansas) Moduli space of G2−instantons on 7−dimensional product manifolds (Huang)
Mar. 13 CANCELED Karin Melnick (University of Maryland) A D'Ambra Theorem in conformal Lorentzian geometry (Dymarz)
Mar. 25 CANCELED Joerg Schuermann (University of Muenster, Germany) An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles (Maxim)
Mar. 27 CANCELED David Massey (Northeastern University) Extracting easily calculable algebraic data from the vanishing cycle complex (Maxim)
Apr. 10 CANCELED Antoine Song (Berkeley) TBA (Chen)

Fall 2019

date speaker title host(s)
Oct. 4 Ruobing Zhang (Stony Brook University) Geometric analysis of collapsing Calabi-Yau spaces (Chen)
Oct. 25 Emily Stark (Utah) Action rigidity for free products of hyperbolic manifold groups (Dymarz)
Nov. 8 Max Forester (University of Oklahoma) Spectral gaps for stable commutator length in some cubulated groups (Dymarz)
Nov. 22 Yu Li (Stony Brook University) On the structure of Ricci shrinkers (Huang)

Spring Abstracts

Xiangdong Xie

The quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces has played an important role in various rigidity questions in geometry and group theory. In these talks I shall give an introduction to this topic. In the first talk I will introduce Gromov hyperbolic spaces, define their ideal boundary, and discuss their basic properties. In the second and third talks I will define the visual metrics on the ideal boundary, explain the connection between quasiisometries of Gromov hyperbolic space and quasiconformal maps on their ideal boundary, and indicate how the quasiconformal structure on the ideal boundary can be used to deduce rigidity.

Kuang-Ru Wu

Following Kobayashi, we consider Griffiths negative complex Finsler bundles, naturally leading us to introduce Griffiths extremal Finsler metrics. As we point out, this notion is closely related to the theory of interpolation of norms, and is characterized by an equation of complex Monge– Ampere type, whose corresponding Dirichlet problem we solve. As applications, we prove that Griffiths extremal Finsler metrics quantize solutions to a natural PDE in Kahler geometry, related to the construction of flat maps for the Mabuchi metric. This is joint work with Tamas Darvas.

Yuanqi Wang

$G_{2}-$instantons are 7-dimensional analogues of flat connections in dimension 3. It is part of Donaldson-Thomas’ program to generalize the fruitful gauge theory in dimensions 2,3,4 to dimensions 6,7,8. The moduli space of $G_{2}-$instantons, with virtual dimension $0$, is expected to have interesting geometric structure and yield enumerative invariant for the underlying $7-$dimensional manifold.

In this talk, in some reasonable special cases and a fairly complete manner, we will describe the relation between the moduli space of $G_{2}-$instantons and an algebraic geometry moduli on a Calabi-Yau 3-fold.

Karin Melnick

D'Ambra proved in 1988 that the isometry group of a compact, simply connected, real-analytic Lorentzian manifold must be compact. I will discuss my recent theorem that the conformal group of such a manifold must also be compact, and how it relates to the Lorentzian Lichnerowicz Conjecture.

Joerg Schuermann

We give an introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles, based on stratified Morse theory for constructible functions. The corresponding local index of an isolated critical point (in a stratified sense) of a one-form depends on the constructible function, specializing for different choices to well known indices like the radial, GSV or Euler obstruction index.

David Massey

Given a complex analytic function on an open subset U of Cn+1, one may consider the complex of sheaves of vanishing cycles along f of the constant sheaf ZU. This complex encodes on the cohomological level the reduced cohomology of the Milnor fibers of f at each of f-1(0). The question is: how does one calculate (ideally, by hand) any useful numbers about this vanishing cycle complex? One answer is to look at the Lê numbers of f. We will discuss the precise relationship between these objects/numbers.

Antoine Song

TBA

Fall Abstracts

Ruobing Zhang

This talk centers on the degenerations of Calabi-Yau metrics. We will focus on the interactions between algebraic degenerations and metric convergence with highly singular behaviors in the collapsing case. As the complex structures degenerate, the collapsing Calabi-Yau metrics may exhibit various wild geometric properties with highly non-algebraic features.

First, as motivating examples, we will describe our recent results on the new collapsing mechanisms of K3 surfaces. Next, we will switch to higher dimensions and we will exhibit some entirely new constructions of degenerating Calabi-Yau metrics which are expected to work in broader contexts. Complex structures degeneration will be accurately characterized by the bubbling and singularity analysis in a geometric manner.

Emily Stark

The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold group is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.

Max Forester

I will discuss stable commutator length (scl) in groups, and some gap theorems for the scl spectrum. Such results say that for various groups, scl of an element is always either zero or is larger than some uniform constant. I will discuss the cases of right-angled Artin groups and certain right-angled Coxeter groups. This is joint work with Pallavi Dani, Ignat Soroko, and Jing Tao.

Yu Li

We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As an application, we show that any Ricci shrinker whose second eigenvalue of the curvature operator is positive must be a quotient of sphere.

Archive of past Geometry seminars

2018-2019 Geometry_and_Topology_Seminar_2018-2019

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

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