Difference between revisions of "Geometry and Topology Seminar"

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== Spring 2017 ==
+
== Fall 2017 ==
  
 
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{| 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| "Free boundary minimal hypersurfaces of Euclidean domains"]]  
+
|[[#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| "The Anomaly Flow and Strominger systems"]]
 
| Lu Wang
 
 
|  
 
|  
 
|-
 
|-
|April 7
+
|December 1
| [https://www.math.rutgers.edu/~feehan/ Paul Feehan] (Rutgers)
+
|TBA
|[[#Paul Feehan| "The Lojasiewicz-Simon gradient inequality and applications to energy discreteness and gradient flows in gauge theory"]]
+
|TBA
| Lu Wang
+
|TBA
 
|-
 
|-
|April 14
+
|December 8
| [https://www.math.wisc.edu/~gong/ Xianghong Gong] (Wisconsin)  
+
|Brian Hepler (Northeastern University)
| [[#Xianghong Gong| "A Frobenius-Nirenberg theorem with parameter"]]
+
|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]
 
|-
 
|
 
 
|}
 
|}
  
== Spring Abstracts ==
+
== Fall Abstracts ==
  
===Lucas Ambrozio===
+
=== Jiyuan Han ===
"Free boundary minimal hypersurfaces of Euclidean domains"
+
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"
  
We will show how the first Betti number of a compact free boundary minimal hypersurface in an domain whose boundary satisfies weak convexity assumptions is controlled effectively by the Morse index of this hypersurface viewed as a critical point of the area functional (joint with A. Carlotto and B. Sharp). Among such domains, the unit three-ball is particularly interesting, as it contains many free boundary minimal surfaces, which one would like to classify. In particular, we will explain how to characterise the critical catenoid in terms of a pinching condition on the second fundamental form (joint with I. Nunes).
+
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.
  
===Paul Feehan===
+
=== Sigurd Angenent ===
"The Lojasiewicz-Simon gradient inequality and applications to energy discreteness and gradient flows in gauge theory"
+
"Topology of closed geodesics on surfaces and curve shortening"
  
The Lojasiewicz-Simon gradient inequality is a generalization, due to Leon Simon (1983), to analytic or Morse-Bott functionals on Banach manifolds of the finite-dimensional gradient inequality, due to Stanislaw Lojasiewicz (1963), for analytic functions on Euclidean space. We shall discuss several recent generalizations of the Lojasiewicz-Simon gradient inequality and a selection of their applications, such as global existence and convergence of Yang-Mills gradient flow over four-dimensional manifolds and discreteness of the energy spectrum for harmonic maps from Riemann surfaces into analytic Riemannian manifolds.
+
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.
  
===Xianghong Gong===
+
=== Ke Zhu===
"A Frobenius-Nirenberg theorem with parameter"
+
"Isometric Embedding via Heat Kernel"
  
The Newlander-Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the complex structure in the complex Euclidean space. We will show two results about the Newlander-Nirenberg theorem with parameter. The first extends the Newlander-Nirenberg theorem to a parametric version, and its proof yields a sharp regularity result as Webster's proof for the Newlander-Nirenberg theorem. The second concerns a version of Nirenberg's complex Frobenius theorem and its proof yields a result with a mild loss of regularity.
+
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.
  
===Rafael Montezuma===
+
=== Shaosai Huang ===
"Metrics of positive scalar curvature and unbounded min-max widths"
+
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"
  
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.
+
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.  
  
===Carmen Rovi===
+
=== Sebastian Baader ===
''The mod 8 signature of a fiber bundle''
+
"A filtration of the Gordian complex via symmetric groups"
  
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.
+
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.
  
===Yair Hartman===
+
=== Shengwen Wang ===
"Intersectional Invariant Random Subgroups and Furstenberg Entropy."
+
"Hausdorff stability of round spheres under small-entropy perturbation"
  
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.
+
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.
All notions will be explained in the talk, no prior knowledge of Invariant Random Subgroups or Furstenberg Entropy is assumed.
+
  
===Bena Tshishiku===
+
=== Marco Mendez-Guaraco ===
 
"TBA"
 
"TBA"
  
===Mark Powell===
+
=== Ovidiu Munteanu ===
''Stable classification of 4-manifolds''
+
"TBA"
 
+
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===
+
"The Anomaly Flow and Strominger systems"
+
 
+
The anomaly flow is a geometric flow which implements the Green-Schwarz anomaly cancellation mechanism originating from superstring theory, while preserving the conformally balanced condition of Hermitian metrics. I will discuss criteria for long time existence and convergence of the flow on toric fibrations with the Fu-Yau ansatz. This is joint work with D.H. Phong and S. Picard.
+
  
 
== Archive of past Geometry seminars ==
 
== Archive of past Geometry seminars ==

Revision as of 13:08, 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