Difference between revisions of "Geometry and Topology Seminar"

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== Spring 2017 ==
+
== Fall 2017 ==
  
 
{| cellpadding="8"
 
{| cellpadding="8"
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!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|Jan 20
+
|September 8
| [http://people.mpim-bonn.mpg.de/rovi/ Carmen Rovi] (University of Indiana Bloomington)
+
|TBA
| [[#Carmen Rovi| "The mod 8 signature of a fiber bundle"]]
+
|TBA
| Maxim
+
|TBA
 
|-
 
|-
|Jan 27
+
|September 15
|
+
|Jiyuan Han (University of Wisconsin-Madison)
|  
+
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]
|  
+
|Local
 
|-
 
|-
|Feb 3
+
|September 22
| Rafael Montezuma (University of Chicago)  
+
|Sigurd Angenent (UW-Madison)
| [[#Rafael Montezuma| "Metrics of positive scalar curvature and unbounded min-max widths"]]
+
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]
| Lu Wang
+
|Local
 
|-
 
|-
|Feb 10
+
|September 29
|
+
|Ke Zhu (Minnesota State University)
|  
+
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]
|  
+
|Bing Wang
 
|-
 
|-
|Feb 17
+
|October 6
|[http://www.math.northwestern.edu/~hartman/ Yair Hartman] (Northwestern University)  
+
|Shaosai Huang (Stony Brook)
|[[#Yair Hartman| "Intersectional Invariant Random Subgroups and Furstenberg Entropy."]]
+
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]
| [http://www.math.wisc.edu/~dymarz Dymarz]
+
|Bing Wang
 
|-
 
|-
|Feb 24
+
|October 13
| [http://math.uchicago.edu/~lambrozio/ Lucas Ambrozio] (University of Chicago)
+
|Sebastian Baader (Bern)
| [[#Lucas Ambrozio| "TBA"]]  
+
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]
| Lu Wang
+
|Kjuchukova
 
|-
 
|-
|March 3
+
|October 20
| [http://www.math.uqam.ca/~powell/ Mark Powell] (Université du Québec à Montréal)
+
|Shengwen Wang (Johns Hopkins)
| [[#Mark Powell| "Stable classification of 4-manifolds"]]
+
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]
| Kjuchukova
+
|Lu Wang
 
|-
 
|-
|March 10
+
|October 27
| [http://www.math.wisc.edu/~kent Autumn Kent] (Wisconsin)
+
|Marco Mendez-Guaraco (Chicago)
| [[#Autumn Kent | ''Analytic functions from hyperbolic manifolds'']]
+
|TBA
| local
+
|Lu Wang
 
|-
 
|-
|March 17
+
|November 3
|  
+
|TBA
|  
+
|TBA
|  
+
|TBA
 +
|-
 +
|November 10
 +
|TBA
 +
|TBA
 +
|TBA
 
|-
 
|-
|March 24
+
|November 17
| Spring Break
+
|Ovidiu Munteanu (University of Connecticut)
 +
|TBA
 +
|Bing Wang
 +
|-
 +
|<b>Thanksgiving Recess</b>
 
|  
 
|  
 
|  
 
|  
|-
 
|March 31
 
| [http://www.math.uci.edu/~xiangwen/ Xiangwen Zhang] (University of California-Irvine)
 
| [[#Xiangwen Zhang| "TBA"]]
 
| Lu Wang
 
 
|  
 
|  
 
|-
 
|-
|April 7
+
|December 1
| reserved
+
|TBA
|
+
|TBA
| Lu Wang
+
|TBA
 
|-
 
|-
|April 14
+
|December 8
| [https://www.math.wisc.edu/~gong/ Xianghong Gong] (Wisconsin)  
+
|Brian Hepler (Northeastern University)
| [[#Xianghong Gong| "TBA"]]
+
|TBA
| local
+
|Max
 
|-
 
|-
|April 21
 
|[http://www.math.csi.cuny.edu/~maher/ Joseph Maher] (CUNY) 
 
| [[#Joseph Maher|"TBA"]]
 
| [http://www.math.wisc.edu/~dymarz Dymarz]
 
|-
 
|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 ===
 
=== Jiyuan Han ===
''Deformation theory of scalar-flat ALE Kahler surfaces''
+
"On closeness of ALE SFK metrics on minimal 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.
+
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK
 +
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.
  
=== Sean Howe ===
+
=== Sigurd Angenent ===
''Representation stability and hypersurface sections''
+
"Topology of closed geodesics on surfaces and curve shortening"
  
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}!
+
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.
  
=== Nan Li ===
+
=== Ke Zhu===
''Quantitative estimates on the singular sets of Alexandrov spaces''
+
"Isometric Embedding via Heat Kernel"
  
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.  
+
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.
  
=== Yu Li ===
+
=== Shaosai Huang ===
 +
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"
  
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.  
+
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.  
  
=== Peyman Morteza ===
+
=== Sebastian Baader ===
''We develop a procedure to construct Einstein metrics by gluing the Calabi metric to an Einstein orbifold.  We show that our gluing problem is obstructed and we calculate the obstruction explicitly.  When our obstruction does not vanish, we obtain a non-existence result in the case that the base orbifold is compact. When our obstruction vanishes and the base orbifold is non-degenerate and asymptotically hyperbolic we prove an existence result. This is a joint work with Jeff Viaclovsky.  ''
+
"A filtration of the Gordian complex via symmetric groups"
  
=== Caglar Uyanik ===
+
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.
''Geometry and dynamics of free group automorphisms''
+
  
A common theme in geometric group theory is to obtain structural results about infinite groups by analyzing their action on metric spaces. In this talk, I will focus on two geometrically significant groups; mapping class groups and outer automorphism groups of free groups.We will describe a particular instance of how the dynamics and geometry of their actions on various spaces provide deeper information about the groups.
+
=== Shengwen Wang ===
 +
"Hausdorff stability of round spheres under small-entropy perturbation"
  
=== Bing Wang ===
+
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.
''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.
+
=== Marco Mendez-Guaraco ===
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 ===
+
''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.
+
 
+
=== Jonathan Zhu ===
+
''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 Zeng===
+
''Short time existence of the Calabi flow with rough initial data''
+
 
+
Calabi flow was introduced by Calabi back in 1950’s as a geometric flow approach to the existence of extremal metrics. Analytically it is a fourth order nonlinear parabolic equation on the Kaehler potentials which deforms the Kaehler potential along its scalar curvature. In this talk, we will show that the Calabi flow admits short time solution for any continuous initial Kaehler metric. This is a joint work with Weiyong He.
+
 
+
== Spring Abstracts ==
+
 
+
===Lucas Ambrozio===
+
 
"TBA"
 
"TBA"
  
===Rafael Montezuma===
+
=== Ovidiu Munteanu ===
"Metrics of positive scalar curvature and unbounded min-max widths"
+
 
+
In this talk, I will construct a sequence of Riemannian metrics on the three-dimensional sphere with scalar curvature greater than or equal to 6, and arbitrarily large min-max widths. The search for such metrics is motivated by a rigidity result of min-max minimal spheres in three-manifolds obtained by Marques and Neves.
+
 
+
===Carmen Rovi===
+
''The mod 8 signature of a fiber bundle''
+
 
+
In this talk we shall be concerned with the residues modulo 4 and modulo 8 of the signature of a 4k-dimensional geometric Poincare complex. I will explain the relation between the signature modulo 8 and two other invariants: the Brown-Kervaire invariant and the Arf invariant. In my thesis I applied the relation between these invariants to the study of the signature modulo 8 of a fiber bundle. In 1973 Werner Meyer used group cohomology to show that a surface bundle has signature divisible by 4. I will discuss current work with David Benson, Caterina Campagnolo and Andrew Ranicki where we are using group cohomology and  representation theory of finite groups to detect non-trivial signatures modulo 8 of surface bundles.
+
 
+
===Yair Hartman===
+
"Intersectional Invariant Random Subgroups and Furstenberg Entropy."
+
 
+
In this talk I'll present a joint work with Ariel Yadin, in which we solve the Furstenberg Entropy Realization Problem for finitely supported random walks (finite range jumps) on free groups and lamplighter groups. This generalizes a previous result of Bowen. The proof consists of several reductions which have geometric and probabilistic flavors of independent interests.
+
All notions will be explained in the talk, no prior knowledge of Invariant Random Subgroups or Furstenberg Entropy is assumed.
+
 
+
===Bena Tshishiku===
+
"TBA"
+
 
+
===Mark Powell===
+
''Stable classification of 4-manifolds''
+
 
+
A stabilisation of a 4-manifold M is a connected sum of M with some number of copies of S^2 x S^2.
+
Two 4-manifolds are said to be stably diffeomorphic if they admit diffeomorphic stabilisations.
+
Since a necessary condition is that the fundamental groups be isomorphic, we study this equivalence relation for a fixed group.  I will discuss recent progress in classifying 4-manifolds up to stable diffeomorphism for certain families of groups, arising from work with Daniel Kasprowski, Markus Land and Peter Teichner. 
+
As a by-product we also obtained a result on the analogous question with the complex projective plane CP^2 replacing S^2 x S^2.
+
 
+
===Autumn Kent===
+
''Analytic functions from hyperbolic manifolds''
+
 
+
At the heart of Thurston's proof of Geometrization for Haken manifolds is a family of analytic functions between Teichmuller spaces called "skinning maps."  These maps carry geometric information about their associated hyperbolic manifolds, and I'll discuss what is presently known about their behavior.  The ideas involved form a mix of geometry, algebra, and analysis.
+
 
+
===Xiangwen Zhang===
+
 
"TBA"
 
"TBA"
  
 
== Archive of past Geometry seminars ==
 
== Archive of past Geometry seminars ==
 +
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>

Revision as of 14:08, 2 October 2017

The Geometry and Topology seminar meets in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm.
For more information, contact Alexandra Kjuchukova or Lu Wang .

Hawk.jpg


Fall 2017

date speaker title host(s)
September 8 TBA TBA TBA
September 15 Jiyuan Han (University of Wisconsin-Madison) "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces" Local
September 22 Sigurd Angenent (UW-Madison) "Topology of closed geodesics on surfaces and curve shortening" Local
September 29 Ke Zhu (Minnesota State University) "Isometric Embedding via Heat Kernel" Bing Wang
October 6 Shaosai Huang (Stony Brook) "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons" Bing Wang
October 13 Sebastian Baader (Bern) "A filtration of the Gordian complex via symmetric groups" Kjuchukova
October 20 Shengwen Wang (Johns Hopkins) "Hausdorff stability of round spheres under small-entropy perturbation" Lu Wang
October 27 Marco Mendez-Guaraco (Chicago) TBA Lu Wang
November 3 TBA TBA TBA
November 10 TBA TBA TBA
November 17 Ovidiu Munteanu (University of Connecticut) TBA Bing Wang
Thanksgiving Recess
December 1 TBA TBA TBA
December 8 Brian Hepler (Northeastern University) TBA Max

Fall Abstracts

Jiyuan Han

"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 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

"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.

Ke Zhu

"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.

Shaosai Huang

"\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.

Sebastian Baader

"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.

Shengwen Wang

"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.

Marco Mendez-Guaraco

"TBA"

Ovidiu Munteanu

"TBA"

Archive of past Geometry seminars

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