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




== Fall 2014==
== Spring 2020 ==
 
 


{| cellpadding="8"
{| cellpadding="8"
Line 16: Line 14:
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|August 29
|Feb. 7
| Yuanqi Wang
|Xiangdong Xie  (Bowling Green University)
| [[#Yuanqi Wang|''Liouville theorem for complex Monge-Ampere equations with conic singularities.'']]
| Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
| [http://www.math.wisc.edu/~bwang Wang]
|(Dymarz)
|-
|-
|September 5
|Feb. 14
|  
|Xiangdong Xie  (Bowling Green University)
|
| Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|
|(Dymarz)
|-
|-
|September 12
|Feb. 21
| [http://people.uwec.edu/daviscw/ Chris Davis] (UW-Eau Claire)
|Xiangdong Xie  (Bowling Green University)
| [[#Chris Davis (UW-Eau Claire)|''L<sup>2</sup> signatures and an example of Cochran-Harvey-Leidy'']]
| Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
| [http://www.math.wisc.edu/~maxim/ Maxim]
|(Dymarz)
|-
|-
|September 19
|Feb. 28
| [http://www.math.northwestern.edu/~knudsen/ Ben Knudsen] (Northwestern)
|Kuang-Ru Wu (Purdue University)
| [[#Ben Knudsen (Northwestern)|''Rational homology of configuration spaces via factorization homology'']]
|Griffiths extremality, interpolation of norms, and Kahler quantization
| [http://www.math.wisc.edu/~ellenber/ Ellenberg]
|(Huang)
|-
|-
|September 26
|Mar. 6
|  
|Yuanqi Wang (University of Kansas)
|
|Moduli space of G2−instantons on 7−dimensional product manifolds
|
|(Huang)
|-
|-
|October 3
|Mar. 13 <b>CANCELED</b>
| [http://homepages.math.uic.edu/~kwhyte/ Kevin Whyte] (UIC)
|Karin Melnick (University of Maryland)
|[[#Kevin Whyte (UIC)|''Quasi-isometric embeddings of symmetric spaces'']]
|A D'Ambra Theorem in conformal Lorentzian geometry
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|(Dymarz)
|-
|-
|October 10
|<b>Mar. 25</b> <b>CANCELED</b>
|[http://math.uchicago.edu/~akwalker/ Alden Walker] (UChicago)
|Joerg Schuermann (University of Muenster, Germany)
|[[#Alden Walker (UChicago)|''Roots, Schottky Semigroups, and a proof of Bandt's Conjecture'']]
|An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|(Maxim)
|-
|-
|October 17
|Mar. 27 <b>CANCELED</b>
|  
|David Massey (Northeastern University)
|
|Extracting easily calculable algebraic data from the vanishing cycle complex
|
|(Maxim)
|-
|-
|October 24
|<b>Apr. 10</b> <b>CANCELED</b>
|  
|Antoine Song (Berkeley)
|
|TBA
|
|(Chen)
|-
|}
|October 31
 
| [http://www2.math.ou.edu/~jing/ Jing Tao] (Oklahoma)
== Fall 2019 ==
| [[#Jing Tao (Oklahoma)|''Growth Tight Actions'']]
 
| [http://www.math.wisc.edu/~rkent/ Kent]
{| cellpadding="8"
!align="left" | date
!align="left" | speaker
!align="left" | title
!align="left" | host(s)
|-
|-
|November 1
|Oct. 4
| [http://www.math.wisc.edu/~dymarz/yggt/ Young Geometric Group Theory in the Midwest Workshop]
|Ruobing Zhang (Stony Brook University)
| Geometric analysis of collapsing Calabi-Yau spaces
|(Chen)
|-
|-
|November 7
| [https://sites.google.com/site/thomasbarthelme/ Thomas Barthelm&eacute;] (Penn State)
| [[#Thomas Barthelm&eacute; (Penn State)|''Counting orbits of Anosov flows in free homotopy classes'']]
| [http://www.math.wisc.edu/~rkent/ Kent]
|-
|-
|November 14
|Oct. 25
| [http://www.math.upenn.edu/~alkju/ Alexandra Kjuchukova] (University of Pennsylvania)
|Emily Stark (Utah)
| [[#Alexandra Kjuchukova (UPenn)|''On the classification of irregular branched covers of four-manifolds'']]
| Action rigidity for free products of hyperbolic manifold groups
| [http://www.math.wisc.edu/~Maxim/ Maxim]
|(Dymarz)
|-
|-
|November 21
|Nov. 8
|  
|Max Forester (University of Oklahoma)
|
|Spectral gaps for stable commutator length in some cubulated groups
|
|(Dymarz)
|-
|-
|Thanksgiving Recess
|Nov. 22
|  
|Yu Li (Stony Brook University)
|
|On the structure of Ricci shrinkers
|
|(Huang)
|-
|-
|December 5
|
|
|
|-
|December 12
| [https://sites.google.com/site/seanlimath/ Sean Li] (UChicago)
|[[#Sean Li (UChicago)|''TBA']]
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|-
|
|}
|}


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


===Yuanqi Wang===
===Xiangdong Xie===
''Liouville theorem for complex Monge-Ampere equations with conic singularities.''
 
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.


Following Calabi, Pogorelov, Evans-Krylov-Safanov, and Trudinger's pioneer work on interior regularities and liouville theorems for Monge-Ampere equations,
===Kuang-Ru Wu===
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)===
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.
''L<sup>2</sup> signatures and an example of Cochran-Harvey-Leidy''


===Ben Knudsen (Northwestern)===
===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.


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


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.
===Karin Melnick===


===Kevin Whyte (UIC)===
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.
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)===
===Joerg Schuermann===
In 1985, Barnsley and Harrington defined a "Mandlebrot set" M for pairs of complex dilations.  This is the set of complex numbers
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.
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.


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.
===David Massey===


===Jing Tao (Oklahoma)===
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.
''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.
===Antoine Song===


===Thomas Barthelm&eacute; (Penn State)===
TBA
''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?
==Fall Abstracts==
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?


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.
===Ruobing Zhang===


===Sean Li (UChicago)===
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"


== Spring 2015 ==
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===


{| cellpadding="8"
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.
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|January 30
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|February 6
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|February 13
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|February 27
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|March 13
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|March 20
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|March 27
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| Spring Break
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|April 10
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| April 17
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|April 24
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== Spring Abstracts ==
===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 ==
== 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]]
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]
<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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