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]]


== Fall 2016 ==
 
== Spring 2020 ==


{| cellpadding="8"
{| cellpadding="8"
Line 13: Line 14:
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|September 9
|Feb. 7
| [http://www.math.wisc.edu/~bwang/ Bing Wang] (UW Madison)
|Xiangdong Xie  (Bowling Green University)
| [[#Bing Wang| "The extension problem of the mean curvature flow"]]
| Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
| (Local)
|(Dymarz)
|-
|-
|September 16
|Feb. 14
| [http://www.math.northwestern.edu/~weinkove/ Ben Weinkove] (Northwestern University)
|Xiangdong Xie  (Bowling Green University)
| [[#Ben Weinkove| "Gauduchon metrics with prescribed volume form"]]
| Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
| Lu Wang
|(Dymarz)
|-
|-
|September 23
|Feb. 21
| Jiyuan Han (UW Madison)
|Xiangdong Xie  (Bowling Green University)
| [[#Jiyuan Han| "Deformation theory of scalar-flat ALE Kahler surfaces"]]
| Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
| (Local)
|(Dymarz)
|-
|-
|September 30
|Feb. 28
|  
|Kuang-Ru Wu (Purdue University)
|  
|Griffiths extremality, interpolation of norms, and Kahler quantization
|
|(Huang)
|-
|-
|October 7
|Mar. 6
| Yu Li (UW Madison)
|Yuanqi Wang (University of Kansas)
| [[#Yu Li| "Ricci flow on asymptotically Euclidean manifolds"]]
|Moduli space of G2−instantons on 7−dimensional product manifolds
| (Local)
|(Huang)
|-
|-
|October 14
|Mar. 13 <b>CANCELED</b>
| [http://math.uchicago.edu/~seanpkh/ Sean Howe] (University of Chicago)
|Karin Melnick (University of Maryland)
| [[#Sean Howe| "Representation stability and hypersurface sections"]]
|A D'Ambra Theorem in conformal Lorentzian geometry
| Melanie Matchett Wood
|(Dymarz)
|-
|-
|October 21
|<b>Mar. 25</b> <b>CANCELED</b>
| [https://sites.google.com/site/mathnanli/ Nan Li] (CUNY)
|Joerg Schuermann (University of Muenster, Germany)
| [[#Nan Li| "Quantitative estimates on the singular Sets of Alexandrov spaces"]]
|An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles
| Lu Wang
|(Maxim)
|-
|-
|October 28
|Mar. 27 <b>CANCELED</b>
| Ronan Conlon
|David Massey (Northeastern University)
| [[#Ronan Conlon| "TBA"]]
|Extracting easily calculable algebraic data from the vanishing cycle complex
| Bing Wang
|(Maxim)
|-
|-
|November 4
|<b>Apr. 10</b> <b>CANCELED</b>
| Jonathan Zhu (Harvard University)
|Antoine Song (Berkeley)
| [[#Jonathan Zhu| "Entropy and self-shrinkers of the mean curvature flow"]]
|TBA
| Lu Wang
|(Chen)
|-
|}
|'''November 7'''
 
| [http://www.massey.ac.nz/massey/expertise/profile.cfm?stref=339830 Gaven Martin] (University of New Zealand)
== Fall 2019 ==
| [[#Gaven Martin| "TBA"]]
 
| Simon Marshall
{| cellpadding="8"
!align="left" | date
!align="left" | speaker
!align="left" | title
!align="left" | host(s)
|-
|-
|November 11
|Oct. 4
| [http://www.math.wisc.edu/~rkent Richard Kent] (Wisconsin)
|Ruobing Zhang (Stony Brook University)
| [[#Richard Kent| ''Analytic functions from hyperbolic manifolds'']]
| Geometric analysis of collapsing Calabi-Yau spaces
| local
|(Chen)
|-
|-
|November 18
| [http://www.math.uiuc.edu/~cuyanik2/ Caglar Uyanik] (Illinois)
| [[#Caglar Uyanik| "TBA"]]
| [http://www.math.wisc.edu/~rkent Kent]
|-
| Thanksgiving Recess
|
|
|
|-
|-
|December 2
|Oct. 25
|Peyman Morteza (UW Madison)
|Emily Stark (Utah)
| [[#Peyman Morteza| "TBA"]]
| Action rigidity for free products of hyperbolic manifold groups
| (Local)
|(Dymarz)
|-
|-
|December 9
|Nov. 8
|
|Max Forester (University of Oklahoma)
|  
|Spectral gaps for stable commutator length in some cubulated groups
|  
|(Dymarz)
|-
|-
|December 16
|Nov. 22
|
|Yu Li (Stony Brook University)
|  
|On the structure of Ricci shrinkers
|  
|(Huang)
|-
|-
|
|}
|}


== 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.


=== Ronan Conlon ===
===Kuang-Ru Wu===
''New examples of gradient expanding K\"ahler-Ricci solitons''


A complete K\"ahler metric $g$ on a K\"ahler manifold $M$ is a \emph{gradient expanding K\"ahler-Ricci soliton} if there exists a smooth real-valued function $f:M\to\mathbb{R}$ with $\nabla^{g}f$ holomorphic such that $\operatorname{Ric}(g)-\operatorname{Hess}(f)+g=0$. I will present new examples of such metrics on the total space of certain holomorphic vector bundles. This is joint work with Alix Deruelle (Universit\'e Paris-Sud).
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.
''Deformation theory of scalar-flat ALE Kahler surfaces''


We prove a Kuranishi-type theorem for deformations of complex structures on ALE Kahler surfaces. This is used to prove that for any scalar-flat Kahler ALE surfaces, all small deformations of complex structure also admit scalar-flat Kahler ALE metrics. A local moduli space of scalar-flat Kahler ALE metrics is then constructed, which is shown to be universal up to small diffeomorphisms (that is, diffeomorphisms which are close to the identity in a suitable sense). A formula for the dimension of the local moduli space is proved in the case of a scalar-flat Kahler ALE surface which deforms to a minimal resolution of \C^2/\Gamma, where \Gamma is a finite subgroup of U(2) without complex reflections. This is a joint work with Jeff Viaclovsky.
===Karin Melnick===


=== Sean Howe ===
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.
''Representation stability and hypersurface sections''


We give stability results for the cohomology of natural local systems on spaces of smooth hypersurface sections as the degree goes to \infty. These results give new geometric examples of a weak version of representation stability for symmetric, symplectic, and orthogonal groups. The stabilization occurs in point-counting and in the Grothendieck ring of Hodge structures, and we give explicit formulas for the limits using a probabilistic interpretation. These results have natural geometric analogs -- for example, we show that the "average" smooth hypersurface in \mathbb{P}^n is \mathbb{P}^{n-1}!
===Joerg Schuermann===
=== Nan Li ===
''Quantitative estimates on the singular sets of Alexandrov spaces''


The definition of quantitative singular sets was initiated by Cheeger and Naber. They proved some volume estimates on such singular sets in non-collapsed manifolds with lower Ricci curvature bounds and their limit spaces. On the quantitative singular sets in Alexandrov spaces, we obtain stronger estimates in a collapsing fashion. We also show that the (k,\epsilon)-singular sets are k-rectifiable and such structure is sharp in some sense. This is a joint work with Aaron Naber.  
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.


=== Yu Li ===
===David Massey===


In this talk, we prove that if an asymptotically Euclidean (AE) manifold with nonnegative scalar curvature has long time existence of Ricci flow, it converges to the Euclidean space in the strong sense. By convergence, the mass will drop to zero as time tends to infinity. Moreover, in three dimensional case, we use Ricci flow with surgery to give an independent proof of positive mass theorem. A classification of diffeomorphism types is also given for all AE 3-manifolds with nonnegative scalar curvature.  
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.


=== Gaven Marin ===
===Antoine Song===
''TBA''


=== Peyman Morteza ===
TBA
''TBA''


=== Richard Kent ===
==Fall Abstracts==
''Analytic functions from hyperbolic manifolds''


Thurston's Geometrization Conjecture, now a celebrated theorem of Perelman, tells us that most 3-manifolds are naturally geometric in nature.  In fact, most 3-manifolds admit hyperbolic metrics.  In the 1970s, Thurston proved the Geometrization conjecture in the case of Haken manifolds, and the proof revolutionized 3-dimensional topology, hyperbolic geometry, Teichm&uuml;ller theory, and dynamics.  Thurston's proof is by induction, constructing a hyperbolic structure from simpler pieces. At the heart of the proof is an analytic function called the ''skinning map'' that one must understand in order to glue hyperbolic structures together.  A better understanding of this map would more brightly illuminate the interaction between topology and geometry in dimension three.  I will discuss what is currently known about this map.
===Ruobing Zhang===


=== Caglar Uyanik ===
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.
''TBA''


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


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


=== Ben Weinkove ===
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.
''Gauduchon metrics with prescribed volume form''


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


=== Jonathan Zhu ===
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.
''Entropy and self-shrinkers of the mean curvature flow''


The Colding-Minicozzi entropy is an important tool for understanding the mean curvature flow (MCF), and is a measure of the complexity of a submanifold. Together with Ilmanen and White, they conjectured that the round sphere minimises entropy amongst all closed hypersurfaces. We will review the basics of MCF and their theory of generic MCF, then describe the resolution of the above conjecture, due to J. Bernstein and L. Wang for dimensions up to six and recently claimed by the speaker for all remaining dimensions. A key ingredient in the latter is the classification of entropy-stable self-shrinkers that may have a small singular set.
===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]]
<br><br>
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]
<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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