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] or [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova].
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 ==


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|<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
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|February 2
|TBA
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|February 9
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|February 16
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|February 23
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|March 2
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|March 9
|TBA
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|March 16
|TBA
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|March 23
|TBA
|TBA
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|<b> Spring Break </b>
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|April 6
|TBA
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|April 13
|TBA
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|April 20
|TBA
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|April 27
|TBA
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|May 4
|TBA
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== Spring Abstracts ==


== Summer Abstracts ==
=== TBA ===


===David Epstein (Warwick)===
TBA
''Splines and manifolds.''
 
[http://www.math.wisc.edu/~rkent/Abstract.Epstein.2015.pdf Abstract (pdf)]
 
-->
 
== Spring 2016 ==
 
Spring 2016: [[Geometry_and_Topology_Seminar_Spring 2016]]
<br><br>
== 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
| [https://uwm.edu/math/people/tran-hung-1/ Hung Tran] (UW Milwaukee)
| [[#Hung Tran|''Relative divergence, subgroup distortion, and geodesic divergence'']]
| [http://www.math.wisc.edu/~dymarz T. Dymarz]
|-
|-
|September 18
|September 15
| [http://www.math.wisc.edu/~dymarz Tullia Dymarz] (UW Madison)
|Jiyuan Han (University of Wisconsin-Madison)
| [[#Tullia Dymarz|''Non-rectifiable Delone sets in amenable groups'']]
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]
| (local)
|Local
|-
|-
|September 25
|September 22
| [https://jpwolfson.wordpress.com/ Jesse Wolfson] (Uchicago)
|Sigurd Angenent (UW-Madison)
| [[#Jesse Wolfson|''Counting Problems and Homological Stability'']]
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]
| [http://www.math.wisc.edu/~mmwood/ M. Matchett Wood]
|Local
|-
|-
|October 2
|September 29
| [https://riemann.unizar.es/~jicogo/ Jose Ignacio Cogolludo Agustín] (University of Zaragoza, Spain)
|Ke Zhu (Minnesota State University)
| [[#Jose Ignacio Cogolludo Agustín|''Topology of curve complements and combinatorial aspects'']]
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]
|[http://www.math.wisc.edu/~maxim L. Maxim]
|Bing Wang
|-
|-
|October 9
|October 6
| [http://people.brandeis.edu/~mcordes/ Matthew Cordes] (Brandeis)
|Shaosai Huang (Stony Brook)
| [[#Matthew Cordes|''Morse boundaries of geodesic metric spaces'']]
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]
| [http://www.math.wisc.edu/~dymarz T. Dymarz]
|Bing Wang
|-
|-
|October 16
|October 13
| [http://www.math.jhu.edu/~bernstein/ Jacob Bernstein] (Johns Hopkins University)
|Sebastian Baader (Bern)
| [[#Jacob Bernstein (Johns Hopkins University)|''Hypersurfaces of low entropy'']]
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]
| [http://www.sites.google.com/a/wisc.edu/lu-wang/ L. Wang]
|Kjuchukova
|-
|-
|October 23
|October 20
| [https://sites.google.com/a/wisc.edu/ysu/ Yun Su] (UW Madison)
|Shengwen Wang (Johns Hopkins)
| [[#Yun Su (Brandeis)|''Higher-order degrees of hypersurface complements.'']]
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]
| (local)
|Lu Wang
|-
|-
|October 30
|October 27
| [http://www.math.stonybrook.edu/phd-student-directory Gao Chen] (Stony Brook University)
|Marco Mendez-Guaraco (Chicago)
| [[#Gao Chen(Stony Brook University)|''Classification of gravitational instantons '']]
|[[#Marco Mendez-Guaraco| "Some geometric aspects of the Allen-Cahn equation"]]
| [http://www.math.wisc.edu/~bwang B.Wang]
|Lu Wang
|-
|-
|November 6
|November 3
| [http://scholar.harvard.edu/gardiner Dan Cristofaro-Gardiner] (Harvard)
|TBA
| [[#Dan Cristofaro-Gardiner|''Higher-dimensional symplectic embeddings and the Fibonacci staircase'']]
|TBA
| [http://www.math.wisc.edu/~kjuchukova Kjuchukova]
|TBA
|
|-
|-
|November 13
|November 10
| [http://people.brandeis.edu/~ruberman/ Danny Ruberman] (Brandeis)
|TBA
| [[#Danny Ruberman|''Configurations of embedded spheres'']]
|TBA
| [http://www.math.wisc.edu/~kjuchukova Kjuchukova]
|TBA
|
|-
|-
|November 20
|November 17
| [https://www.math.toronto.edu/cms/izosimov-anton/ Anton Izosimov] (University of Toronto)
|Ovidiu Munteanu (University of Connecticut)
| [[#Anton Izosimov (University of Toronto)|''TBA'']]
|[[#Ovidiu Munteanu| "The geometry of four dimensional shrinking Ricci solitons"]]
| [http://www.math.wisc.edu/~maribeff/ Mari-Beffa]
|Bing Wang
|-
|-
|Thanksgiving Recess
|<b>Thanksgiving Recess</b>
|
|
|  
|  
|
|
|-
|-
|December 4
|December 1
| [http://www.math.wisc.edu/~westrich/ Quinton Westrich] (UW Madison)
|TBA
| [[#Quinton Westrich (UW Madison) |''Harmonic Chern Forms on Polarized Kähler Manifolds'']]
|TBA
| (local)
|TBA
|-
|-
|December 11
|December 8
|[http://kaihowong.weebly.com/ Tommy Wong] (UW Madison)
|Brian Hepler (Northeastern University)
| [[#Tommy Wong (UW Madison)|''Milnor Fiber of Complex Hyperplane Arrangement.'']]
|[[#Brian Hepler| "Deformation Formulas for Parameterizable Hypersurfaces"]]
| (local)
|Max
|-
|-
|
|}
|}


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


=== Jiyuan Han ===
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"


===Hung Tran===
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK
''Relative divergence, subgroup distortion, and geodesic divergence''
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.


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
=== Sigurd Angenent ===
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 $r^s$.
"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.


===Tullia Dymarz===
=== Ke Zhu===
''Non-rectifiable Delone sets in amenable groups''
"Isometric Embedding via Heat Kernel"


In 1998 Burago-Kleiner and McMullen constructed the first
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.
examples of coarsely dense and uniformly discrete subsets of R^n that are
not biLipschitz equivalent to the standard lattice Z^n. Similarly we
find subsets inside the three dimensional solvable Lie group SOL that are
not bilipschitz to any lattice in SOL. The techniques involve combining
ideas from Burago-Kleiner with quasi-isometric rigidity results from
geometric group theory.


===Jesse Wolfson===
=== Shaosai Huang ===
''Counting Problems and Homological Stability''
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"


In 1969, Arnold showed that the i^{th} homology of the space of un-ordered configurations of n points in the plane becomes independent of n for n>>i. A decade later, Segal extended Arnold's method to show that the i^{th} homology of the space of degree n holomorphic maps from \mathbb{P}^1 to itself also becomes independent of n for large n, and, moreover, that both sequences of spaces have the same limiting homology. We explain how, using Weil's number field/function field dictionary, one might have predicted this topological coincidence from easily verifiable statements about specific counting problems.  We then discuss ongoing joint work with Benson Farb and Melanie Wood in which we use other counting problems to predict and discover new instances of homological stability in the topology of complex manifolds.  
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"


===Matthew Cordes===
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.
''Morse boundaries of geodesic metric spaces''


I will introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with equivalence classes of geodesic rays that identify the ``hyperbolic directions" in that space. (A ray is Morse if quasi-geodesics with endpoints on the ray stay bounded distance from the ray.) This boundary is a quasi-isometry invariant and a visibility space. In the case of a proper CAT(0) space the Morse boundary generalizes the contracting boundary of Charney and Sultan and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. Time permitting I will also discuss some results on Morse boundary of the mapping class group and briefly describe joint work with David Hume developing a capacity dimension for the Morse boundary.
=== Shengwen Wang ===
"Hausdorff stability of round spheres under small-entropy perturbation"


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


===Jacob Bernstein===
=== Marco Mendez-Guaraco ===
''Hypersurfaces of low entropy''
"Some geometric aspects of the Allen-Cahn equation"


The entropy is a quantity introduced by Colding and Minicozzi and may be thought of as a rough measure of the geometric complexity of a hypersurface of Euclidean space.  It is closely related to the mean curvature flow.  On the one hand, the entropy controls the dynamics of the flow. On the other hand, the mean curvature flow may be used to study the entropy. In this talk I will survey some recent results with Lu Wang that show that hypersurfaces of low entropy really are simple.
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.


===Yun Su===
=== Ovidiu Munteanu ===
''Higher-order degrees of hypersurface complements.''
"The geometry of four dimensional shrinking Ricci solitons"


===Gao Chen===
I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons.
''Classification of gravitational instantons''


A gravitational instanton is a noncompact complete hyperkahler manifold of real dimension 4 with faster than quadratic curvature decay. In this talk, I will discuss the recent work towards the classification of gravitational instantons. This is a joint work with X. X. Chen.
=== Brian Hepler ===
"Deformation Formulas for Parameterizable Hypersurfaces"


===Dan Cristofaro-Gardiner===
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 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.
''Higher-dimensional symplectic embeddings and the Fibonacci staircase''
 
McDuff and Schlenk determined when a four dimensional symplectic ellipsoid can be embedded into a ball, and found that when the ellipsoid is close to round, the answer is given by an infinite staircase determined by the odd-index Fibonacci numbers.  I will explain joint work with Richard Hind, showing that a generalization of this holds in all even dimensions.
 
===Danny Ruberman===
''Configurations of embedded spheres''
 
Configurations of lines in the plane have been studied since antiquity. In recent years, combinatorial methods have been used to decide if a specified incidence relation between certain objects ("lines") and other objects ("points") can be realized by actual points and lines in a projective plane over a field. For the real and complex fields, one can weaken the condition to look for topologically embedded lines (circles in the real case, spheres in the complex case) that meet according to a specified incidence relation. I will explain some joint work with Laura Starkston (Stanford) giving new topological restrictions on the realization of configurations of spheres in the complex projective plane.


== 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