Geometry and Topology Seminar 2019-2020: Difference between revisions

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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 ==
== Spring 2020 ==


{| cellpadding="8"
{| cellpadding="8"
Line 14: Line 14:
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|January 26
|Feb. 7
|Jingrui Cheng
|Xiangdong Xie  (Bowling Green University)
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence"]]
| Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|Local
|(Dymarz)
|-
|-
|February 2
|Feb. 14
|Jingrui Cheng
|Xiangdong Xie  (Bowling Green University)
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence" (continued)]]
| Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|Local
|(Dymarz)
|-
|-
|February 9
|Feb. 21
|Jingrui Cheng
|Xiangdong Xie  (Bowling Green University)
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence" (continued)]]
| Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|Local
|(Dymarz)
|-
|-
|February 16
|Feb. 28
|TBA
|Kuang-Ru Wu (Purdue University)
|TBA
|Griffiths extremality, interpolation of norms, and Kahler quantization
|TBA
|(Huang)
|-
|February 23
|TBA
|TBA
|TBA
|-
|March 2
|TBA
|TBA
|TBA
|-
|March 9
|TBA
|TBA
|TBA
|-
|March 16
|Yu Li
|[[#Yu Li|"The Rigidity of Ricci shrinkers of dimension four"]]
|Bing Wang
|-
|March 23
|TBA
|TBA
|TBA
|-
|<b> Spring Break </b>
|
|
|-
|-
|April 6
|Mar. 6
|Wei Ho
|Yuanqi Wang (University of Kansas)
|TBA
|Moduli space of G2−instantons on 7−dimensional product manifolds
|Daniel Erman
|(Huang)
|-
|-
|April 13
|Mar. 13 <b>CANCELED</b>
|TBA
|Karin Melnick (University of Maryland)
|TBA
|A D'Ambra Theorem in conformal Lorentzian geometry
|TBA
|(Dymarz)
|-
|-
|April 20
|<b>Mar. 25</b> <b>CANCELED</b>
|Pei-Ken Hung (Columbia Univ.)
|Joerg Schuermann (University of Muenster, Germany)
|TBA
|An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles
|Lu Wang
|(Maxim)
|-
|-
|April 27
|Mar. 27 <b>CANCELED</b>
|TBA
|David Massey (Northeastern University)
|TBA
|Extracting easily calculable algebraic data from the vanishing cycle complex
|TBA
|(Maxim)
|-
|-
|May 4
|<b>Apr. 10</b> <b>CANCELED</b>
|TBA
|Antoine Song (Berkeley)
|TBA
|TBA
|TBA
|-
|(Chen)
|
|}
|}


== Spring Abstracts ==
== Fall 2019 ==
 
=== 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"
{| cellpadding="8"
Line 108: Line 68:
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|September 8
|Oct. 4
|TBA
|Ruobing Zhang (Stony Brook University)
|TBA
| Geometric analysis of collapsing Calabi-Yau spaces
|TBA
|(Chen)
|-
|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
|Oct. 25
|Shengwen Wang (Johns Hopkins)
|Emily Stark (Utah)
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]
| Action rigidity for free products of hyperbolic manifold groups
|Lu Wang
|(Dymarz)
|-
|-
|October 27
|Nov. 8
|Marco Mendez-Guaraco (Chicago)
|Max Forester (University of Oklahoma)
|[[#Marco Mendez-Guaraco| "Some geometric aspects of the Allen-Cahn equation"]]
|Spectral gaps for stable commutator length in some cubulated groups
|Lu Wang
|(Dymarz)
|-
|-
|November 3
|Nov. 22
|TBA
|Yu Li (Stony Brook University)
|TBA
|On the structure of Ricci shrinkers
|TBA
|(Huang)
|-
|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 ==
==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.


=== Jiyuan Han ===
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.
"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
===Karin Melnick===
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 ===
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.
"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.
===Joerg Schuermann===


=== Ke Zhu===
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.
"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.
===David Massey===


=== Shaosai Huang ===
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.
"\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.
===Antoine Song===


=== Sebastian Baader ===
TBA
"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.
==Fall Abstracts==


=== Shengwen Wang ===
===Ruobing Zhang===
"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.
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.


=== Marco Mendez-Guaraco ===
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.
"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.
===Emily Stark===


=== Ovidiu Munteanu ===
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.
"The geometry of four dimensional shrinking Ricci solitons"


I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons.
===Max Forester===


=== Brian Hepler ===
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.
"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.
===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 ==
== Archive of past Geometry seminars ==
2018-2019  [[Geometry_and_Topology_Seminar_2018-2019]]
<br><br>
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 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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