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/~rkent Richard Kent].
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]]


== Spring 2015 ==


== Spring 2018 ==


{| cellpadding="8"
{| cellpadding="8"
Line 14: Line 14:
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|January 23
|January 26
|  
|TBA
|
|TBA
|
|TBA
|-
|-
|January 30
|February 2
|  
|TBA
|
|TBA
|
|TBA
|-
|-
|February 6
|February 9
| [http://www.math.wisc.edu/~strenner/ Balazs Strenner] (Wisconsin)
|TBA
| [[#Balazs Strenner (Wisconsin) |''Penner’s conjecture on pseudo-Anosov mapping classes.'']]
|TBA
| local
|TBA
|-
|-
|<b>Thursday, February 12, at 11AM in VV 901</b>
|February 16
| [http://rybu.org/ Ryan Budney] (Victoria)
|TBA
| [[#Ryan Budney (Victoria)|''Operads and spaces of knots.'']]
|TBA
| [http://www.math.wisc.edu/~rkent/ Kent]
|TBA
|-
|-
|February 20
|February 23
| [http://www.math.illinois.edu/~jsapir2/ Jenya Sapir] (UIUC)
|TBA
|[[#Jenya Sapir (UIUC) |''Counting non-simple closed curves on surfaces.'']]
|TBA
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|TBA
|-
|-
|February 27
|March 2
|  
|TBA
|
|TBA
|
|TBA
|-
|-
|March 6
|March 9
|[http://www3.nd.edu/~bwang3/ Botong Wang] (Notre Dame)
|TBA
|[[#Botong Wang (Notre Dame) |''Deformation theory with cohomology constraints.'']]
|TBA
|Max
|TBA
|
|-
|-
|March 13
|March 16
| [http://www.math.vanderbilt.edu/~saleaw/ Andrew Sale] (Vanderbilt)
|TBA
|[[#Andrew Sale (Vanderbilt) | "TBA"]]
|TBA
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|TBA
|-
|-
|March 20
|March 23
|
|TBA
|
|TBA
|
|TBA
|-
|-
|March 27
|<b> Spring Break </b>
|
|
|
|
|
|-
|-
| Spring Break
|April 6
|  
|TBA
|
|TBA
|
|TBA
|-
|-
|April 10
|April 13
|| [http://homepages.math.uic.edu/~mbhull/ Michael Hull] (UIC)
|TBA
|[[#Michael Hull (UIC)|''TBA'']]
|TBA
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|TBA
|-
|-
| April 17
|April 20
|| [https://sites.google.com/site/seanlimath/ Sean Li] (UChicago)
|TBA
|[[#Sean Li (UChicago)|''Coarse differentiation of Lipschitz functions.'']]
|TBA
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|TBA
|-
|-
|April 24
|April 27
|  
|TBA
|
|TBA
|
|TBA
|-
|-
|May 1
|May 4
|  
|TBA
|
|TBA
|
|TBA
|-
|-
|May 8
|
|
|
|
|-
|}
|}
== Spring Abstracts ==
== Spring Abstracts ==


===Balazs Strenner (Wisconsin)===
=== TBA ===
''Penner’s conjecture on pseudo-Anosov mapping classes.''
 
There are many constructions of pseudo-Anosov elements of mapping class groups of surfaces. Some of them are known to generate all pseudo-Anosov mapping classes, others are known not to. In 1988, Penner gave a very general construction of pseudo-Anosov mapping classes, and he conjectured that all pseudo-Anosov mapping classes arise this way up to finite power. This conjecture was known to be true on some simple surfaces, including the torus, but has otherwise remained open. In this talk I prove that the conjecture is false for most surfaces. (This  is joint work with Hyunshik Shin.)
 
===Ryan Budney (Victoria)===
''Operads and spaces of knots.''
 
I will describe a connection between the geometrization of 3-manifolds and a subject called operads.  It manifests itself as a structure theorem for the space of smooth embeddings of the circle in the 3-sphere and points to a homotopy-theoretic approach to old questions about finite-type invariants.
 
===Jenya  Sapir (UIUC)===
''Counting non-simple closed curves on surfaces.''
 
We show how to get coarse bounds on the number of (non-simple) closed geodesics on a surface, given upper bounds on both length and self-intersection number. Recent work by Mirzakhani has produced asymptotics for the growth of the number of simple closed curves, and this work has been extended to curves with at most 3 self-intersections with respect to length. However, no asymptotics, or even bounds, were previously known for other bounds on self-intersection number. Time permitting, we will discuss some applications of this result.
 
===Botong Wang (Notre Dame)===
''Deformation theory with cohomology constraints.''
 
Deformation theory is a powerful tool to study the local structure of moduli spaces.  I will first give an introduction to the theory of Deligne-Goldman-Millson, which translates deformation theory problems to problems of differential graded Lie algebras. I will also talk about a generalization to deformation theory problems with cohomology constraints. This is used to study the local structure of cohomology jump loci in various moduli spaces.


===Andrew Sale (Vanderbilt)===
TBA
"TBA"
 
===Michael Hull (UIC)===
"TBA"
 
===Sean Li (UChicago)===
''Coarse differentiation of Lipschitz functions.''
 
Bates, Johnson, Lindenstrauss, Preiss, and Schechtman introduced a notion of large scale differentiation for Lipschitz functions between normed linear spaces.  We discuss an extension of this result to the nonabelian setting of Carnot groups and use it to derive quantitative estimates for nonembeddability of such groups into certain classes of metric spaces.
 
== Fall 2014==




== Fall 2017 ==


{| cellpadding="8"
{| cellpadding="8"
Line 140: Line 105:
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|August 29
|September 8
| Yuanqi Wang
|TBA
| [[#Yuanqi Wang|''Liouville theorem for complex Monge-Ampere equations with conic singularities.'']]
|TBA
| [http://www.math.wisc.edu/~bwang Wang]
|TBA
|-
|-
|September 5
|September 15
|  
|Jiyuan Han (University of Wisconsin-Madison)
|
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]
|
|Local
|-
|-
|September 12
|September 22
| [http://people.uwec.edu/daviscw/ Chris Davis] (UW-Eau Claire)
|Sigurd Angenent (UW-Madison)
| [[#Chris Davis (UW-Eau Claire)|''L<sup>2</sup> signatures and an example of Cochran-Harvey-Leidy'']]
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]
| [http://www.math.wisc.edu/~maxim/ Maxim]
|Local
|-
|-
|September 19
|September 29
| [http://www.math.northwestern.edu/~knudsen/ Ben Knudsen] (Northwestern)
|Ke Zhu (Minnesota State University)
| [[#Ben Knudsen (Northwestern)|''Rational homology of configuration spaces via factorization homology'']]
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]
| [http://www.math.wisc.edu/~ellenber/ Ellenberg]
|Bing Wang
|-
|-
|September 26
|October 6
|
|Shaosai Huang (Stony Brook)
|
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]
|
|Bing Wang
|-
|October 3
| [http://homepages.math.uic.edu/~kwhyte/ Kevin Whyte] (UIC)
|[[#Kevin Whyte (UIC)|''Quasi-isometric embeddings of symmetric spaces'']]
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|-
|October 10
|[http://math.uchicago.edu/~akwalker/ Alden Walker] (UChicago)
|[[#Alden Walker (UChicago)|''Roots, Schottky Semigroups, and a proof of Bandt's Conjecture'']]
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|-
|-
|October 17
|October 13
|  
|Sebastian Baader (Bern)
|
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]
|
|Kjuchukova
|-
|-
|October 24
|October 20
|  
|Shengwen Wang (Johns Hopkins)
|
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]
|
|Lu Wang
|-
|-
|October 31
|October 27
| [http://www2.math.ou.edu/~jing/ Jing Tao] (Oklahoma)
|Marco Mendez-Guaraco (Chicago)
| [[#Jing Tao (Oklahoma)|''Growth Tight Actions'']]
|[[#Marco Mendez-Guaraco| "Some geometric aspects of the Allen-Cahn equation"]]
| [http://www.math.wisc.edu/~rkent/ Kent]
|Lu Wang
|-
|-
|November 1
|November 3
| [http://www.math.wisc.edu/~dymarz/yggt/ Young Geometric Group Theory in the Midwest Workshop]
|TBA
|TBA
|TBA
|-
|-
|November 7
|November 10
| [https://sites.google.com/site/thomasbarthelme/ Thomas Barthelm&eacute;] (Penn State)
|TBA
| [[#Thomas Barthelm&eacute; (Penn State)|''Counting orbits of Anosov flows in free homotopy classes'']]
|TBA
| [http://www.math.wisc.edu/~rkent/ Kent]
|TBA
|-
|-
|November 14
|November 17
| [http://www.math.upenn.edu/~alkju/ Alexandra Kjuchukova] (University of Pennsylvania)
|Ovidiu Munteanu (University of Connecticut)
| [[#Alexandra Kjuchukova (University of Pennsylvania)|''On the classification of irregular branched covers of four-manifolds'']]
|[[#Ovidiu Munteanu| "The geometry of four dimensional shrinking Ricci solitons"]]
| [http://www.math.wisc.edu/~Maxim/ Maxim]
|Bing Wang
|-
|-
|November 21
|<b>Thanksgiving Recess</b>
|
|  
|  
|
|
|-
|Thanksgiving Recess
|  
|  
|
|
|-
|-
|December 4, <b>Thursday at 4pm in VV 901</b>
|December 1
| Oyku Yurttas (Georgia Tech)
|TBA
|[[#Oyku Yurttas (Georgia Tech)|''Dynnikov and train track transition matrices of pseudo-Anosov braids'']]
|TBA
|[http://www.math.wisc.edu/~jeanluc/ Thiffeault]
|TBA
|-
|-
|December 5
|December 8
| No seminar.
|Brian Hepler (Northeastern University)
|
|[[#Brian Hepler| "Deformation Formulas for Parameterizable Hypersurfaces"]]
|
|Max
|-
|December 12
| No seminar.
|
|
|-
|-
|
|}
|}


== Fall Abstracts ==
== Fall Abstracts ==


===Yuanqi Wang===
=== Jiyuan Han ===
''Liouville theorem for complex Monge-Ampere equations with conic singularities.''
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"
 
Following Calabi, Pogorelov, Evans-Krylov-Safanov, and Trudinger's pioneer work on interior regularities and liouville theorems for Monge-Ampere equations,
we prove the  Liouville theorem for conic K&auml;hler-Ricci flat metrics. We also discuss various applications of this Liouville theorem to conic K&auml;hler geometry.


===Chris Davis (UW-Eau Claire)===
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK
''L<sup>2</sup> signatures and an example of Cochran-Harvey-Leidy''
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.


===Ben Knudsen (Northwestern)===
=== Sigurd Angenent ===
"Topology of closed geodesics on surfaces and curve shortening"


''Rational homology of configuration spaces via factorization homology''
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.


The study of configuration spaces is particularly tractable over a field of characteristic zero, and much effort has gone into producing chain complexes simple enough for explicit computations, formulas for Betti numbers, and homological stability results. I will discuss recent work identifying the homology of the configuration spaces of an arbitrary manifold M with the homology of a certain Lie algebra constructed from the compactly supported cohomology of M. The aforementioned results follow immediately from this identification, albeit with hypotheses removed; in particular, one obtains a new, elementary proof of homological stability for configuration spaces.
=== Ke Zhu===
"Isometric Embedding via Heat Kernel"


===Kevin Whyte (UIC)===
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.
The rigidity theorems of Mostow and Margulis for lattices in semi simple Lie groups are some of the most celebrated in their field, and are motivation for much of geometric group theory.     Mostow's result, which states that every isomorphism between lattices extends to an equivariant isometry between symmetric spaces, has been generalized by Kleiner and Leeb to say that any map between higher rank symmetric spaces which is quasi-isometric (a large scale version of bilipschitz) is actually a perturbation of an isometry.    Margulis' superrigidity theorem, which says every homomorphism between lattices which has infinite image extends to an equivariant isometric embedding of symmetric spaces, has resisted a generalization of this sort.   We will discuss one such result, which considers when quasi-isometric embeddings of symmetric spaces are near isometric embeddings.    Our results show that the situation is complicated - in some cases one does have rigidity while in others there are exotic quasi-isometric embeddings.    What geometric properties distinguish the two cases is only starting to be understood.   This is joint work with David Fisher (Indiana).


===Alden Walker (UChicago)===
=== Shaosai Huang ===
In 1985, Barnsley and Harrington defined a "Mandlebrot set" M for pairs of complex dilations.  This is the set of complex numbers
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"
c such that the limit set generated by the pair of dilations x-> cx+1 and x-> cx-1 is connected.  The set M is also the closure of the set of roots of polynomials with coefficients in {-1,0,1}.  As with the usual Mandlebrot set, M has strong connections to dynamics and algebra, and it has been studied by Bousch, Bandt, Solomyak, Xu, Thurston, and Tiozzo. For a geometric group theorist, the study of M is qualitatively similar to the study of Kleinian groups acting on their limit sets or on universal circles.


Barnsley and Harrington noted the (numerically apparent) existence of infinitely many "holes" in M, which correspond to exotic components of the space of Schottky semigroups. Bandt rigorously confirmed a single hole in 2002 and conjectured that the interior of M is dense in M away from the real axis. We give the new technique of "traps" to certify an interior point of M, and we use these traps to prove Bandt's conjecture and certify the existence of infinitely many holes in M.
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.  


The only prerequisite for this talk is point-set topology.  Fun pictures will be provided.  This is joint work with Danny Calegari and Sarah Koch.
=== Sebastian Baader ===
"A filtration of the Gordian complex via symmetric groups"


===Jing Tao (Oklahoma)===
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.
''Growth Tight Actions''


Let G be a group equipped with a finite generating set S. G is called growth tight if its exponential growth rate relative to S is strictly greater than that of every quotient G/N with N infinite. This notion was first introduced by Grigorchuk and de la Harpe. Examples of groups that are growth tight include free groups relative to bases and, more generally, hyperbolic groups relative to any generating set. In this talk, I will provide some sufficient conditions for growth tightness which encompass all previous known examples.
=== Shengwen Wang ===
"Hausdorff stability of round spheres under small-entropy perturbation"


===Thomas Barthelm&eacute; (Penn State)===
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.
''Counting orbits of Anosov flows in free homotopy classes''


In 1972, Plante and Thurston asked the following question: If M is a manifold supporting an Anosov flow, does the number of conjugacy classes in the fundamental group grows exponentially fast with the length of the shortest orbit representative?
=== Marco Mendez-Guaraco ===
This question can be solved by answering the following: Can one give an upper bound on the growth rate of the length of orbits inside a free homotopy class?
"Some geometric aspects of the Allen-Cahn equation"


In this talk, I will explain how one can use the geometry and topology of Anosov flows to answer both questions in the 3-manifold case. This is joint work with Sergio Fenley.
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.


===Alexandra Kjuchukova (University of Pennsylvania)===
=== Ovidiu Munteanu ===
''On the classification of irregular branched covers of four-manifolds''
"The geometry of four dimensional shrinking Ricci solitons"


It is a famous result of Hilden and Montesinos that every closed orientable three-manifold can be realized as an irregular three-fold cover of S^3 branched over a knot. In contrast, a smooth four-manifold which can be realized as a cover of S^4 branched over a smooth surface must have signature equal to zero. Given two simply-connected, closed, oriented four-manifolds X and Y and a surface B embedded in X with an isolated singularity, I will prove a necessary condition for the existence of an irregular dihedral branched covering map f: Y -> X with branching set B. Conversely, given a simply-connected oriented closed four-manifold X, I will outline a construction realizing as irregular dihedral covers of X infinitely many (and conjecturally all) of the manifolds Y afforded by the necessary condition.
I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons.  


===Oyku Yurttas (Georgia Tech)===
=== Brian Hepler ===
''Dynnikov and train track transition matrices of pseudo-Anosov braids''
"Deformation Formulas for Parameterizable Hypersurfaces"


In this talk we will compare a <i>Dynnikov matrix</i> with the train track transition matrix of a given pseudo-Anosov braid on the finitely punctured disk. Our main result is that these matrices are isospectral up to roots of unity and some zeros under particular conditions.
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]]
<br><br>
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]
<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