Difference between revisions of "Geometry and Topology Seminar"

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The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.
 
The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.
 
<br>  
 
<br>  
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .
+
For more information, contact Shaosai Huang.
  
 
[[Image:Hawk.jpg|thumb|300px]]
 
[[Image:Hawk.jpg|thumb|300px]]
  
== Fall 2016 ==
 
  
{| cellpadding="8"
 
!align="left" | date
 
!align="left" | speaker
 
!align="left" | title
 
!align="left" | host(s)
 
|-
 
|September 9
 
| [http://www.math.wisc.edu/~bwang/ Bing Wang] (UW Madison)
 
| [[#Bing Wang| "The extension problem of the mean curvature flow"]]
 
| (Local)
 
|-
 
|September 16
 
| [http://www.math.northwestern.edu/~weinkove/ Ben Weinkove] (Northwestern University)
 
| [[#Ben Weinkove| "Gauduchon metrics with prescribed volume form"]]
 
| Lu Wang
 
|-
 
|September 23
 
| Jiyuan Han (UW Madison)
 
| [[#Jiyuan Han| "Deformation theory of scalar-flat ALE Kahler surfaces"]]
 
| (Local)
 
|-
 
|September 30
 
|
 
|
 
|
 
|-
 
|October 7
 
| Yu Li (UW Madison) 
 
| [[#Yu Li| "Ricci flow on asymptotically Euclidean manifolds"]]
 
| (Local)
 
|-
 
|October 14
 
| [http://math.uchicago.edu/~seanpkh/ Sean Howe] (University of Chicago)
 
| [[#Sean Howe| "Representation stability and hypersurface sections"]]
 
| Melanie Matchett Wood
 
|-
 
|October 21
 
| [https://sites.google.com/site/mathnanli/ Nan Li] (CUNY) 
 
| [[#Nan Li| "Quantitative estimates on the singular Sets of Alexandrov spaces"]]
 
| Lu Wang
 
|-
 
|October 28
 
| Ronan Conlon(Florida International University)
 
| [[#Ronan Conlon| "New examples of gradient expanding K\"ahler-Ricci solitons"]]
 
| Bing Wang
 
|-
 
|November 4
 
| Jonathan Zhu (Harvard University)
 
| [[#Jonathan Zhu| "Entropy and self-shrinkers of the mean curvature flow"]]
 
| Lu Wang
 
|-
 
|November 11
 
|  [http://www.math.wisc.edu/~rkent Richard Kent] (Wisconsin)
 
| [[#Richard Kent| ''Analytic functions from hyperbolic manifolds'']]
 
| local
 
|-
 
|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
 
|Peyman Morteza (UW Madison)
 
| [[#Peyman Morteza| "TBA"]]
 
| (Local) 
 
|-
 
|December 9
 
| Yu Zeng(University of Rochester)
 
|  [[#Yu Zeng| "TBA"]]
 
| Bing Wang
 
|
 
|-
 
|December 16
 
|
 
|
 
|-
 
|
 
|}
 
  
== Spring 2017 ==
+
 
 +
== Fall 2019 ==
  
 
{| cellpadding="8"
 
{| cellpadding="8"
Line 99: Line 16:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|Jan 20
+
|Oct. 4
+
|Ruobing Zhang (Stony Brook University)
|
+
| Geometric analysis of collapsing Calabi-Yau spaces
|
+
|(Chen)
|-
 
|Jan 27
 
 
|
 
|
 
|-
 
|Feb 3
 
 
|
 
|
 
|-
 
|Feb 10
 
 
|
 
|
 
|-
 
|Feb 17
 
 
|
 
|
 
|-
 
|Feb 24
 
 
|
 
|  
 
|-
 
|March 3
 
 
|
 
|
 
|-
 
|March 10
 
 
|
 
|
 
|-
 
|March 17
 
 
|
 
|
 
|-
 
|March 24
 
|  Spring Break
 
|
 
|
 
 
|-
 
|-
|March 31
 
 
|
 
|
 
 
|-
 
|-
|April 7
+
|Oct. 25
|
+
|Emily Stark (Utah)
|  
+
| Action rigidity for free products of hyperbolic manifold groups
|  
+
|(Dymarz)
 
|-
 
|-
|April 14
+
|Nov. 8
|
+
|Max Forester (University of Oklahoma)
|  
+
|Spectral gaps for stable commutator length in some cubulated groups
|  
+
|(Dymarz)
 
|-
 
|-
|April 21
+
|Nov. 22
|
+
|Yu Li (Stony Brook University)
|  
+
|On the structure of Ricci shrinkers
|  
+
|(Huang)
 
|-
 
|-
|April 28
 
| [http://bena-tshishiku.squarespace.com/ Bena Tshishiku] (Harvard)
 
| [[#Bena Tshishiku| "TBA"]]
 
| [http://www.math.wisc.edu/~dymarz Dymarz]
 
|-
 
|
 
 
|}
 
|}
  
== Fall Abstracts ==
+
==Fall Abstracts==
 
 
=== Ronan Conlon ===
 
''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).
 
 
 
 
 
=== Jiyuan Han ===
 
''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.
 
 
 
=== Sean Howe ===
 
''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}!
 
=== 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.
 
 
 
=== Yu Li ===
 
 
 
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.
 
 
 
=== Gaven Marin ===
 
''TBA''
 
 
 
=== Peyman Morteza ===
 
''TBA''
 
 
 
=== Richard Kent ===
 
''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.
 
 
 
=== Caglar Uyanik ===
 
''TBA''
 
 
 
=== Bing Wang ===
 
''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.
+
===Ruobing Zhang===
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 ===
+
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.
''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.
+
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.
  
=== Jonathan Zhu ===
+
===Emily Stark===
''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.
+
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===
  
== Spring Abstracts ==
+
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.
  
===Bena Tshishiku===
+
===Yu Li===
"TBA"
+
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 08:44, 4 November 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



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)

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