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/~dymarz Tullia Dymarz].
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .


[[Image:Hawk.jpg|thumb|300px]]
[[Image:Hawk.jpg|thumb|300px]]


== Summer 2015 ==


== Spring 2018 ==


{| cellpadding="8"
{| cellpadding="8"
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!align="left" | host(s)
!align="left" | host(s)
|-
|-
|<b>June 23 at 2pm in Van Vleck 901</b>
|January 26
| [http://www2.warwick.ac.uk/fac/sci/maths/people/staff/david_epstein/ David Epstein] (Warwick)
|TBA
| [[#David Epstein (Warwick) |''Splines and manifolds.'']]
|TBA
| Hirsch
|TBA
|-
|-
|February 2
|TBA
|TBA
|TBA
|-
|February 9
|TBA
|TBA
|TBA
|-
|February 16
|TBA
|TBA
|TBA
|-
|February 23
|TBA
|TBA
|TBA
|-
|March 2
|TBA
|TBA
|TBA
|-
|March 9
|TBA
|TBA
|TBA
|-
|March 16
|TBA
|TBA
|TBA
|-
|March 23
|TBA
|TBA
|TBA
|-
|<b> Spring Break </b>
|
|
|-
|April 6
|TBA
|TBA
|TBA
|-
|April 13
|TBA
|TBA
|TBA
|-
|April 20
|TBA
|TBA
|TBA
|-
|April 27
|TBA
|TBA
|TBA
|-
|May 4
|TBA
|TBA
|TBA
|-
|
|}
|}
== Spring Abstracts ==


== Summer Abstracts ==
=== TBA ===
 
===David Epstein (Warwick)===
''Splines and manifolds.''
 
[http://www.math.wisc.edu/~rkent/Abstract.Epstein.2015.pdf Abstract (pdf)]


TBA




== Fall 2015==
== Fall 2017 ==
 
 


{| cellpadding="8"
{| cellpadding="8"
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!align="left" | host(s)
!align="left" | host(s)
|-
|-
|September 4
|September 8
|  
|TBA
|
|TBA
|
|TBA
|-
|-
|September 11
|September 15
| [Hung Tran] (UW Milwaukee)
|Jiyuan Han (University of Wisconsin-Madison)
| [[#Hung Tran (UW Milwaukee)|''Relative divergence, subgroup distortion, and geodesic divergence'']]
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]
| [http://www.math.wisc.edu/~dymarz T. Dymarz]
|Local
|-
|-
|September 18
|September 22
|  
|Sigurd Angenent (UW-Madison)
|
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]
|
|Local
|-
|-
|September 25
|September 29
|  
|Ke Zhu (Minnesota State University)
|
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]
|
|Bing Wang
|-
|-
|October 2
|October 6
|  
|Shaosai Huang (Stony Brook)
|
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]
|
|Bing Wang
|-
|-
|October 9
|October 13
|  
|Sebastian Baader (Bern)
|
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]
|
|Kjuchukova
|-
|-
|October 16
|October 20
| [http://www.math.jhu.edu/~bernstein/ Jacob Bernstein] (Johns Hopkins University)
|Shengwen Wang (Johns Hopkins)
| [[#Jacob Bernstein (Johns Hopkins University)|''TBA'']]
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]
| [http://math.jhu.edu/~lwang/ L. Wang]
|Lu Wang
|-
|-
|October 23
|October 27
| [https://www.math.toronto.edu/cms/izosimov-anton/ Anton Izosimov] (University of Toronto)
|Marco Mendez-Guaraco (Chicago)
| [[#Anton Izosimov (University of Toronto)|''TBA'']]
|[[#Marco Mendez-Guaraco| "Some geometric aspects of the Allen-Cahn equation"]]
| [http://www.math.wisc.edu/~maribeff/ Mari-Beffa]
|Lu Wang
|-
|-
|October 30
|November 3
|
|TBA
|
|TBA
|
|TBA
|-
|-
|November 6
|November 10
|
|TBA
|
|TBA
|
|TBA
|-
|-
|November 13
|November 17
|
|Ovidiu Munteanu (University of Connecticut)
|
|[[#Ovidiu Munteanu| "The geometry of four dimensional shrinking Ricci solitons"]]
|
|Bing Wang
|-
|-
|November 20
|<b>Thanksgiving Recess</b>
|
|  
|  
|
|
|-
|Thanksgiving Recess
|  
|  
|
|
|-
|-
|December 4
|December 1
|
|TBA
|
|TBA
|
|TBA
|-
|-
|December 11
|December 8
|
|Brian Hepler (Northeastern University)
|
|[[#Brian Hepler| "Deformation Formulas for Parameterizable Hypersurfaces"]]
|
|Max
|-
|-
|
|}
|}


== Fall Abstracts ==
== 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"


===Hung Tran===
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.
''Relative divergence, subgroup distortion, and geodesic divergence''


In my presentation, I introduce three new invariants for pairs $(G;H)$ consisting of a finitely generated group $G$ and a subgroup $H$. The first invariant is the upper relative divergence which generalizes Gersten's notion of divergence. The second invariant is the lower relative divergence which generalizes a definition of Cooper-Mihalik. The third invariant is the lower subgroup distortion which parallels the standard notion
=== Sebastian Baader ===
of subgroup distortion. We examine the relative divergence (both upper and lower) of a group with respect to a normal subgroup or a cyclic subgroup. We also explore relative divergence of $CAT(0)$ groups and relatively hyperbolic groups with respect to various subgroups to better understand geometric properties of these groups. We answer the question of Behrstock and Drutu about the existence of Morse geodesics in $CAT(0)$ spaces with divergence function strictly greater than $r^n$ and strictly less than $r^{n+1}$, where $n$ is an integer greater than $1$. More precisely, we show that for each real number $s>2$, there is a $CAT(0)$ space $X$ with a proper and cocompact action of some finitely generated group such that $X$ contains a Morse bi-infinite geodesic with the divergence equivalent to $r^s$.
"A filtration of the Gordian complex via symmetric groups"


===Anton Izosimov===
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.
''TBA''


===Jacob Bernstein===
=== Shengwen Wang ===
''TBA''
"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 ===
"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.
 
=== Ovidiu Munteanu ===
"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.
 
=== Brian Hepler ===
"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.


== 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]]
<br><br>
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]
<br><br>
<br><br>

Revision as of 02:58, 12 January 2018

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


Spring 2018

date speaker title host(s)
January 26 TBA TBA TBA
February 2 TBA TBA TBA
February 9 TBA TBA TBA
February 16 TBA TBA TBA
February 23 TBA TBA TBA
March 2 TBA TBA TBA
March 9 TBA TBA TBA
March 16 TBA TBA TBA
March 23 TBA TBA TBA
Spring Break
April 6 TBA TBA TBA
April 13 TBA TBA TBA
April 20 TBA TBA TBA
April 27 TBA TBA TBA
May 4 TBA TBA TBA

Spring Abstracts

TBA

TBA


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) "Some geometric aspects of the Allen-Cahn equation" Lu Wang
November 3 TBA TBA TBA
November 10 TBA TBA TBA
November 17 Ovidiu Munteanu (University of Connecticut) "The geometry of four dimensional shrinking Ricci solitons" Bing Wang
Thanksgiving Recess
December 1 TBA TBA TBA
December 8 Brian Hepler (Northeastern University) "Deformation Formulas for Parameterizable Hypersurfaces" 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

"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.

Ovidiu Munteanu

"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.

Brian Hepler

"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.

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