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 Shaosai Huang.


 
[[Image:Hawk.jpg|thumb|300px]]
== Fall 2012==




== Spring 2020 ==


{| cellpadding="8"
{| cellpadding="8"
Line 14: Line 14:
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|September 21
|Feb. 7
| [http://www.math.wisc.edu/~josizemore/ Owen Sizemore] (Wisconsin)
|Xiangdong Xie  (Bowling Green University)
| [[#Owen Sizemore (Wisconsin) |
| Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
''Operator Algebra Techniques in Measureable Group Theory'']]
|(Dymarz)
| local
|-
|-
|September 28
|Feb. 14
|[https://engineering.purdue.edu/~mboutin/ Mireille Boutin] (Purdue)
|Xiangdong Xie  (Bowling Green University)
|[[#Mireille Boutin (Purdue) |
| Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
''The Pascal Triangle of a discrete Image: <br>
|(Dymarz)
definition, properties, and application to object segmentation'']]
|[http://www.math.wisc.edu/~maribeff/ Mari Beffa]
|-
|-
|October 5
|Feb. 21
| [http://www.math.msu.edu/~schmidt/ Ben Schmidt] (Michigan State)
|Xiangdong Xie  (Bowling Green University)
| [[#Ben Schmidt (Michigan State)|
| Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
''Three manifolds of constant vector curvature'']]
|(Dymarz)
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|-
|-
|October 12
|Feb. 28
| [https://www2.bc.edu/ian-p-biringer/ Ian Biringer] (Boston College)
|Kuang-Ru Wu (Purdue University)
| [[#Ian Biringer (Boston College)|
|Griffiths extremality, interpolation of norms, and Kahler quantization
''Growth of Betti numbers and a probabilistic take on Gromov Hausdorff convergence'']]
|(Huang)
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|-
|-
|October 19
|Mar. 6
| Peng Gao (Simons Center for Geometry and Physics)
|Yuanqi Wang (University of Kansas)
| [[#Peng Gao (Simons Center for Geometry and Physics)|
|Moduli space of G2−instantons on 7−dimensional product manifolds
''string theory partition functions and geodesic spectrum'']]
|(Huang)
|[http://www.math.wisc.edu/~bwang/ Wang]
|-
|-
|October 26
|Mar. 13 <b>CANCELED</b>
| [http://www.math.wisc.edu/~nelson/ Jo Nelson] (Wisconsin)
|Karin Melnick (University of Maryland)
| [[#Jo Nelson (Wisconsin) |
|A D'Ambra Theorem in conformal Lorentzian geometry
''Cylindrical contact homology as a well-defined homology theory? Part I'']]
|(Dymarz)
| local
|-
|-
|November 2
|<b>Mar. 25</b> <b>CANCELED</b>
| [http://www.bowdoin.edu/~jtaback/ Jennifer Taback] (Bowdoin)
|Joerg Schuermann (University of Muenster, Germany)
| [[#Jennifer Taback (Bowdoin)|
|An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles
''TBA'']]
|(Maxim)
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|-
|-
|November 9
|Mar. 27 <b>CANCELED</b>
| [http://math.uchicago.edu/~wilsonj/ Jenny Wilson] (Chicago)
|David Massey (Northeastern University)
| [[#Jenny Wilson (Chicago)|
|Extracting easily calculable algebraic data from the vanishing cycle complex
''TBA'']]
|(Maxim)
| [http://www.math.wisc.edu/~ellenber/ Ellenberg]
|-
|-
|November 16
|<b>Apr. 10</b> <b>CANCELED</b>
|[http://www.math.uic.edu/people/profile?id=GasJ574 Jonah Gaster] (UIC)
|Antoine Song (Berkeley)
|[[#Jonah Gaster (UIC)|
|TBA
''TBA'']]
|(Chen)
|[http://www.math.wisc.edu/~rkent/ Kent]
|}
 
== Fall 2019 ==
 
{| cellpadding="8"
!align="left" | date
!align="left" | speaker
!align="left" | title
!align="left" | host(s)
|-
|-
| Thanksgiving Recess
|Oct. 4
|
|Ruobing Zhang (Stony Brook University)
|
| Geometric analysis of collapsing Calabi-Yau spaces
|
|(Chen)
|-
|-
|November 30
| [http://www.its.caltech.edu/~shinpei/ Shinpei Baba] (Caltech)
|[[#Shinpei Baba (Caltech)|
''TBA'']]
|[http://www.math.wisc.edu/~rkent/ Kent]
|-
|-
|December 7
|Oct. 25
| [http://math.uchicago.edu/~mann/ Kathryn Mann] (Chicago)
|Emily Stark (Utah)
|[[#Kathryn Mann (Chicago)|
| Action rigidity for free products of hyperbolic manifold groups
''TBA'']]
|(Dymarz)
|[http://www.math.wisc.edu/~rkent/ Kent]
|-
|-
|December 14
|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)
|-
|-
|}
|}


== Fall Abstracts ==
==Spring Abstracts==


===Owen Sizemore (Wisconsin)===
===Xiangdong Xie===
''Operator Algebra Techniques in Measureable Group Theory''


Measurable group theory is the study of groups via their actions on measure spaces. While the classification for amenable groups was essentially complete by the early 1980's, progress for nonamenable groups has been slow to emerge. The last 15 years has seen a surge in the classification of ergodic actions of nonamenable groups, with methods coming from diverse areas. We will survey these new results, as well as, give an introduction to the operator algebra techniques that have been used.
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.


===Mireille Boutin (Purdue)===
===Kuang-Ru Wu===
''The Pascal Triangle of a discrete Image: definition, properties, and application to object segmentation''


We define the Pascal Triangle of a discrete (gray scale) image as a pyramidal ar-
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.
rangement of complex-valued moments and we explore its geometric significance. In
particular, we show that the entries of row k of this triangle correspond to the Fourier
series coefficients of the moment of order k of the Radon transform of the image. Group
actions on the plane can be naturally prolonged onto the entries of the Pascal Triangle. We study the induced action of some common group actions, such as translation,
rotations, and reflections, and we propose simple tests for equivalence and self-
equivalence for these group actions. The motivating application of this work is the
problem of recognizing ”shapes” on images, for example characters, digits or simple
graphics. Application to the MERGE project, in which we developed a fast method for segmenting hazardous material signs on a cellular phone, will be also discussed.  


This is joint work with my graduate students Shanshan Huang and Andrew Haddad.
===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.  


===Ben Schmidt (Michigan State)===
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.
''Three manifolds of constant vector curvature.''


A Riemannian manifold M is said to have extremal curvature K if all sectional curvatures are bounded above by K or if all sectional curvatures are bounded below by K.  A manifold with extremal curvature K has constant vector curvature K if every tangent vector to M belongs to a tangent plane of curvature K.  For surfaces, having constant vector curvature is equivalent to having constant curvature.  In dimension three, the eight Thurston geometries all have constant vector curvature.  In this talk, I will discuss the classification of closed three manifolds with constant vector curvature.  Based on joint work with Jon Wolfson.
===Karin Melnick===


===Ian Biringer (Boston College)===
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.
''Growth of Betti numbers and a probabilistic take on Gromov Hausdorff convergence''


We will describe an asymptotic relationship between the volume and the Betti numbers of certain locally symmetric spaces. The proof uses an exciting new tool: a synthesis of Gromov-Hausdorff convergence of Riemannian manifolds and Benjamini-Schramm convergence from graph theory.
===Joerg Schuermann===


===Peng Gao (Simons Center for Geometry and Physics)===
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.
''string theory partition functions and geodesic spectrum''


String theory partition functions often have nice modular properties, which is well understood within the context of representation theory of (supersymmetric extensions) of Virasoro algebra.
===David Massey===
However, many questions of physical importance are preferrably addressed when string theory is formulated in terms of non-linear sigma model on a Riemann surface with a Riemannian manifold as target space. Traditionally, physicists have studied such sigma models within the realm of perturbation theory, overlooking a large class of very natural critical points of the path integral, namely, closed geodesics on the target space Riemannian manifold. We propose how to take into account the effect of these critical points on the path integral, and initiate its study on Ricci flat targe spaces, such as the K3 surface.


===Jo Nelson (Wisconsin)===
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.
''Cylindrical contact homology as a well-defined homology theory? Part I''


In this talk I will define all the concepts in the title, starting with what a contact manifold is.  I will also  explain how the heuristic arguments sketched in the literature since 1999 fail to define a homology theory and provide a foundation for a well-defined cylindrical contact homology, while still providing an invariant of the contact structure.  A later talk will provide us with a large class of examples under which one can compute a well-defined version of cylindrical contact homology via a new approach the speaker developed for her thesis that is distinct and completely independent of previous specialized attempts.
===Antoine Song===


===Jennifer Taback (Bowdoin)===
TBA
''TBA''


===Jenny Wilson (Chicago)===
==Fall Abstracts==
''TBA''


===Jonah Gaster (UIC)===
===Ruobing Zhang===
''TBA''


===Shinpei Baba (Caltech)===
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.
''TBA''


===Kathryn Mann (Chicago)===
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''


== Spring 2013 ==
===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.


{| cellpadding="8"
===Max Forester===
!align="left" | date
!align="left" | speaker
!align="left" | title
!align="left" | host(s)
|-
|January 25
|
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|February 1
|
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|-
|February 8
|
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|February 15
|
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|February 22
|
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|March 1
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|March 8
|
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|March 15
|
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|-
|March 22
| [http://www-personal.umich.edu/~mishlie/ Michelle Lee] (Michigan)
|[[#Michelle Lee (Michigan)|
''TBA'']]
|[http://www.math.wisc.edu/~rkent/ Kent]
|-
|Spring Break
|
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|April 5
|
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|April 12
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|April 19
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|April 26
|
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|May 3
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|May 10
|
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|}


== Spring Abstracts ==
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.


===Michelle Lee (Michigan)===
===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]]
<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]]
<br><br>
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]
<br><br>
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]
<br><br>
<br><br>
2010: [[Fall-2010-Geometry-Topology]]
2010: [[Fall-2010-Geometry-Topology]]

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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