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/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .
For more information, contact Shaosai Huang.


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




 
== Spring 2020 ==
== Spring 2017 ==


{| cellpadding="8"
{| cellpadding="8"
Line 15: Line 14:
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|Jan 20
|Feb. 7
| [http://people.mpim-bonn.mpg.de/rovi/ Carmen Rovi] (University of Indiana Bloomington)
|Xiangdong Xie  (Bowling Green University)
| [[#Carmen Rovi| "The mod 8 signature of a fiber bundle"]]
| Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
| Maxim
|(Dymarz)
|-
|-
|Jan 27
|Feb. 14
|   
|Xiangdong Xie (Bowling Green University)
|  
| Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|  
|(Dymarz)
|-
|-
|Feb 3
|Feb. 21
| Rafael Montezuma (University of Chicago)  
|Xiangdong Xie  (Bowling Green University)
| [[#Rafael Montezuma| "Metrics of positive scalar curvature and unbounded min-max widths"]]
| Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
| Lu Wang
|(Dymarz)
|-
|-
|Feb 10
|Feb. 28
|
|Kuang-Ru Wu (Purdue University)
|  
|Griffiths extremality, interpolation of norms, and Kahler quantization
|  
|(Huang)
|-
|-
|Feb 17
|Mar. 6
|[http://www.math.northwestern.edu/~hartman/ Yair Hartman] (Northwestern University)  
|Yuanqi Wang (University of Kansas)
|[[#Yair Hartman| "Intersectional Invariant Random Subgroups and Furstenberg Entropy."]]
|Moduli space of G2−instantons on 7−dimensional product manifolds
| [http://www.math.wisc.edu/~dymarz Dymarz]
|(Huang)
|-
|-
|Feb 24
|Mar. 13 <b>CANCELED</b>
| [http://math.uchicago.edu/~lambrozio/ Lucas Ambrozio] (University of Chicago)
|Karin Melnick (University of Maryland)
| [[#Lucas Ambrozio| "Free boundary minimal hypersurfaces of Euclidean domains"]]
|A D'Ambra Theorem in conformal Lorentzian geometry
| Lu Wang
|(Dymarz)
|-
|-
|March 3
|<b>Mar. 25</b> <b>CANCELED</b>
| [http://www.math.uqam.ca/~powell/ Mark Powell] (Université du Québec à Montréal)
|Joerg Schuermann (University of Muenster, Germany)
| [[#Mark Powell| "Stable classification of 4-manifolds"]]
|An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles
| Kjuchukova
|(Maxim)
|-
|-
|March 10
|Mar. 27 <b>CANCELED</b>
| [http://www.math.wisc.edu/~kent Autumn Kent] (Wisconsin)
|David Massey (Northeastern University)
| [[#Autumn Kent | ''Analytic functions from hyperbolic manifolds'']]
|Extracting easily calculable algebraic data from the vanishing cycle complex
| local
|(Maxim)
|-
|-
|March 17
|<b>Apr. 10</b> <b>CANCELED</b>
|  
|Antoine Song (Berkeley)
|  
|TBA
|  
|(Chen)
|-
|}
|March 24
 
| Spring Break
== Fall 2019 ==
|  
 
|  
{| cellpadding="8"
!align="left" | date
!align="left" | speaker
!align="left" | title
!align="left" | host(s)
|-
|-
|March 31
|Oct. 4
| [http://www.math.uci.edu/~xiangwen/ Xiangwen Zhang] (University of California-Irvine)
|Ruobing Zhang (Stony Brook University)
| [[#Xiangwen Zhang| "TBA"]]
| Geometric analysis of collapsing Calabi-Yau spaces
| Lu Wang
|(Chen)
|  
|-
|-
|April 7
| [https://www.math.rutgers.edu/~feehan/ Paul Feehan] (Rutgers)
|[[#Paul Feehan| "TBA"]]
| Lu Wang
|-
|-
|April 14
|Oct. 25
| [https://www.math.wisc.edu/~gong/ Xianghong Gong] (Wisconsin)  
|Emily Stark (Utah)
| [[#Xianghong Gong| "TBA"]]
| Action rigidity for free products of hyperbolic manifold groups
| local
|(Dymarz)
|-
|-
|April 21
|Nov. 8
|[http://www.math.csi.cuny.edu/~maher/ Joseph Maher] (CUNY)
|Max Forester (University of Oklahoma)
| [[#Joseph Maher|"TBA"]]
|Spectral gaps for stable commutator length in some cubulated groups
| [http://www.math.wisc.edu/~dymarz Dymarz]
|(Dymarz)
|-
|-
|April 28
|Nov. 22
| [http://bena-tshishiku.squarespace.com/ Bena Tshishiku] (Harvard)
|Yu Li (Stony Brook University)
| [[#Bena Tshishiku| "TBA"]]
|On the structure of Ricci shrinkers
| [http://www.math.wisc.edu/~dymarz Dymarz]
|(Huang)
|-
|-
|
|}
|}


== Spring Abstracts ==
==Spring Abstracts==
 
===Xiangdong Xie===
 
The quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces has played
an important role in various  rigidity questions in geometry and group theory.
In these talks I  shall give an introduction to this topic.  In the first talk I will introduce Gromov hyperbolic spaces, define their ideal boundary,  and  discuss their basic properties.  In the second and third talks I will define the visual metrics on the ideal boundary, explain the connection between quasiisometries of  Gromov hyperbolic space and quasiconformal maps on  their ideal boundary, and indicate  how the quasiconformal structure on the ideal boundary can be used to deduce rigidity.
 
===Kuang-Ru Wu===
 
Following Kobayashi, we consider Griffiths negative complex Finsler bundles, naturally leading us to introduce Griffiths extremal Finsler metrics. As we point out, this notion is closely related to the theory of interpolation of norms, and is characterized by an equation of complex Monge– Ampere type, whose corresponding Dirichlet problem we solve. As applications, we prove that Griffiths extremal Finsler metrics quantize solutions to a natural PDE in Kahler geometry, related to the construction of flat maps for the Mabuchi metric. This is joint work with Tamas Darvas.
 
===Yuanqi Wang===
$G_{2}-$instantons are 7-dimensional analogues of flat connections in dimension 3. It is part of Donaldson-Thomas’ program to generalize the fruitful gauge theory in dimensions 2,3,4 to dimensions 6,7,8. The moduli space of  $G_{2}-$instantons, with virtual dimension $0$, is  expected to have interesting  geometric structure and  yield enumerative invariant for the underlying $7-$dimensional manifold.
 
In this talk, in some reasonable special cases and a fairly complete manner, we will describe the relation between the moduli space of $G_{2}-$instantons and an algebraic geometry moduli on a Calabi-Yau 3-fold.
 
===Karin Melnick===
 
D'Ambra proved in 1988 that the isometry group of a compact, simply connected, real-analytic Lorentzian manifold must be compact. I will discuss my recent theorem that the conformal group of such a manifold must also be compact, and how it relates to the Lorentzian Lichnerowicz Conjecture.


===Lucas Ambrozio===
===Joerg Schuermann===
"Free boundary minimal hypersurfaces of Euclidean domains"


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).
We give an introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles, based on stratified Morse theory for constructible functions. The corresponding local index of an isolated critical point (in a stratified sense) of a one-form depends on the constructible function, specializing for different choices to well known indices like the radial, GSV or Euler obstruction index.


===Paul Feehan===
===David Massey===
"TBA"


===Rafael Montezuma===
Given a complex analytic function on an open subset U  of C<sup>n+1</sup>, one may consider the complex of sheaves of vanishing cycles along f of the constant sheaf Z<sub>U</sub>. This complex encodes on the cohomological level the reduced cohomology of the Milnor fibers of f at each of f<sup>-1</sup>(0). The question is: how does one calculate (ideally, by hand)  any useful numbers about this vanishing cycle complex? One answer is to look at the Lê numbers of f. We will discuss the precise relationship between these objects/numbers.
"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.
===Antoine Song===


===Carmen Rovi===
TBA
''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.
==Fall Abstracts==


===Yair Hartman===
===Ruobing Zhang===
"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.
This talk centers on the degenerations of Calabi-Yau metrics. We will focus on the interactions between algebraic degenerations and metric convergence with highly singular behaviors in the collapsing case. As the complex structures degenerate, the collapsing Calabi-Yau metrics may exhibit various wild geometric properties with highly non-algebraic features.
All notions will be explained in the talk, no prior knowledge of Invariant Random Subgroups or Furstenberg Entropy is assumed.


===Bena Tshishiku===
First, as motivating examples, we will describe our recent results on the new collapsing mechanisms of K3 surfaces. Next, we will switch to higher dimensions and we will exhibit some entirely new constructions of degenerating Calabi-Yau metrics which are expected to work in broader contexts. Complex structures degeneration will be accurately characterized by the bubbling and singularity analysis in a geometric manner.
"TBA"


===Mark Powell===
===Emily Stark===
''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.
The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold group is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.
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===
===Max Forester===
''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.
I will discuss stable commutator length (scl) in groups, and some gap theorems for the scl spectrum. Such results say that for various groups, scl of an element is always either zero or is larger than some uniform constant. I will discuss the cases of right-angled Artin groups and certain right-angled Coxeter groups. This is joint work with Pallavi Dani, Ignat Soroko, and Jing Tao.


===Xiangwen Zhang===
===Yu Li===
"TBA"
We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As an application, we show that any Ricci shrinker whose second eigenvalue of the curvature operator is positive must be a quotient of sphere.


== Archive of past Geometry seminars ==
== Archive of past Geometry seminars ==
2018-2019  [[Geometry_and_Topology_Seminar_2018-2019]]
<br><br>
2017-2018 [[Geometry_and_Topology_Seminar_2017-2018]]
<br><br>
2016-2017  [[Geometry_and_Topology_Seminar_2016-2017]]
2016-2017  [[Geometry_and_Topology_Seminar_2016-2017]]
<br><br>
<br><br>

Latest revision as of 18:56, 3 September 2020

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

Hawk.jpg


Spring 2020

date speaker title host(s)
Feb. 7 Xiangdong Xie (Bowling Green University) Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces (Dymarz)
Feb. 14 Xiangdong Xie (Bowling Green University) Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces (Dymarz)
Feb. 21 Xiangdong Xie (Bowling Green University) Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces (Dymarz)
Feb. 28 Kuang-Ru Wu (Purdue University) Griffiths extremality, interpolation of norms, and Kahler quantization (Huang)
Mar. 6 Yuanqi Wang (University of Kansas) Moduli space of G2−instantons on 7−dimensional product manifolds (Huang)
Mar. 13 CANCELED Karin Melnick (University of Maryland) A D'Ambra Theorem in conformal Lorentzian geometry (Dymarz)
Mar. 25 CANCELED Joerg Schuermann (University of Muenster, Germany) An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles (Maxim)
Mar. 27 CANCELED David Massey (Northeastern University) Extracting easily calculable algebraic data from the vanishing cycle complex (Maxim)
Apr. 10 CANCELED Antoine Song (Berkeley) TBA (Chen)

Fall 2019

date speaker title host(s)
Oct. 4 Ruobing Zhang (Stony Brook University) Geometric analysis of collapsing Calabi-Yau spaces (Chen)
Oct. 25 Emily Stark (Utah) Action rigidity for free products of hyperbolic manifold groups (Dymarz)
Nov. 8 Max Forester (University of Oklahoma) Spectral gaps for stable commutator length in some cubulated groups (Dymarz)
Nov. 22 Yu Li (Stony Brook University) On the structure of Ricci shrinkers (Huang)

Spring Abstracts

Xiangdong Xie

The quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces has played an important role in various rigidity questions in geometry and group theory. In these talks I shall give an introduction to this topic. In the first talk I will introduce Gromov hyperbolic spaces, define their ideal boundary, and discuss their basic properties. In the second and third talks I will define the visual metrics on the ideal boundary, explain the connection between quasiisometries of Gromov hyperbolic space and quasiconformal maps on their ideal boundary, and indicate how the quasiconformal structure on the ideal boundary can be used to deduce rigidity.

Kuang-Ru Wu

Following Kobayashi, we consider Griffiths negative complex Finsler bundles, naturally leading us to introduce Griffiths extremal Finsler metrics. As we point out, this notion is closely related to the theory of interpolation of norms, and is characterized by an equation of complex Monge– Ampere type, whose corresponding Dirichlet problem we solve. As applications, we prove that Griffiths extremal Finsler metrics quantize solutions to a natural PDE in Kahler geometry, related to the construction of flat maps for the Mabuchi metric. This is joint work with Tamas Darvas.

Yuanqi Wang

$G_{2}-$instantons are 7-dimensional analogues of flat connections in dimension 3. It is part of Donaldson-Thomas’ program to generalize the fruitful gauge theory in dimensions 2,3,4 to dimensions 6,7,8. The moduli space of $G_{2}-$instantons, with virtual dimension $0$, is expected to have interesting geometric structure and yield enumerative invariant for the underlying $7-$dimensional manifold.

In this talk, in some reasonable special cases and a fairly complete manner, we will describe the relation between the moduli space of $G_{2}-$instantons and an algebraic geometry moduli on a Calabi-Yau 3-fold.

Karin Melnick

D'Ambra proved in 1988 that the isometry group of a compact, simply connected, real-analytic Lorentzian manifold must be compact. I will discuss my recent theorem that the conformal group of such a manifold must also be compact, and how it relates to the Lorentzian Lichnerowicz Conjecture.

Joerg Schuermann

We give an introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles, based on stratified Morse theory for constructible functions. The corresponding local index of an isolated critical point (in a stratified sense) of a one-form depends on the constructible function, specializing for different choices to well known indices like the radial, GSV or Euler obstruction index.

David Massey

Given a complex analytic function on an open subset U of Cn+1, one may consider the complex of sheaves of vanishing cycles along f of the constant sheaf ZU. This complex encodes on the cohomological level the reduced cohomology of the Milnor fibers of f at each of f-1(0). The question is: how does one calculate (ideally, by hand) any useful numbers about this vanishing cycle complex? One answer is to look at the Lê numbers of f. We will discuss the precise relationship between these objects/numbers.

Antoine Song

TBA

Fall Abstracts

Ruobing Zhang

This talk centers on the degenerations of Calabi-Yau metrics. We will focus on the interactions between algebraic degenerations and metric convergence with highly singular behaviors in the collapsing case. As the complex structures degenerate, the collapsing Calabi-Yau metrics may exhibit various wild geometric properties with highly non-algebraic features.

First, as motivating examples, we will describe our recent results on the new collapsing mechanisms of K3 surfaces. Next, we will switch to higher dimensions and we will exhibit some entirely new constructions of degenerating Calabi-Yau metrics which are expected to work in broader contexts. Complex structures degeneration will be accurately characterized by the bubbling and singularity analysis in a geometric manner.

Emily Stark

The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold group is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.

Max Forester

I will discuss stable commutator length (scl) in groups, and some gap theorems for the scl spectrum. Such results say that for various groups, scl of an element is always either zero or is larger than some uniform constant. I will discuss the cases of right-angled Artin groups and certain right-angled Coxeter groups. This is joint work with Pallavi Dani, Ignat Soroko, and Jing Tao.

Yu Li

We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As an application, we show that any Ricci shrinker whose second eigenvalue of the curvature operator is positive must be a quotient of sphere.

Archive of past Geometry seminars

2018-2019 Geometry_and_Topology_Seminar_2018-2019

2017-2018 Geometry_and_Topology_Seminar_2017-2018

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

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