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]]


== Spring 2015 ==


== Spring 2020 ==


{| cellpadding="8"
{| cellpadding="8"
Line 14: Line 14:
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|January 23
|Feb. 7
|  
|Xiangdong Xie  (Bowling Green University)
|
| Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|
|(Dymarz)
|-
|-
|January 30
|Feb. 14
|  
|Xiangdong Xie  (Bowling Green University)
|
| Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|
|(Dymarz)
|-
|-
|February 6
|Feb. 21
| [http://www.math.wisc.edu/~strenner/ Balazs Strenner] (Wisconsin)
|Xiangdong Xie  (Bowling Green University)
| [[#Balazs Strenner (Wisconsin) |''Penner’s conjecture on pseudo-Anosov mapping classes.'']]
| Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
| local
|(Dymarz)
|-
|-
|<b>Thursday, February 12, at 11AM in VV 901</b>
|Feb. 28
| [http://rybu.org/ Ryan Budney] (Victoria)
|Kuang-Ru Wu (Purdue University)
| [[#Ryan Budney (Victoria)|''Operads and spaces of knots.'']]
|Griffiths extremality, interpolation of norms, and Kahler quantization
| [http://www.math.wisc.edu/~rkent/ Kent]
|(Huang)
|-
|-
|February 20
|Mar. 6
| [http://www.math.illinois.edu/~jsapir2/ Jenya Sapir] (UIUC)
|Yuanqi Wang (University of Kansas)
|[[#Jenya Sapir (UIUC) |''Counting non-simple closed curves on surfaces.'']]
|Moduli space of G2−instantons on 7−dimensional product manifolds
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|(Huang)
|-
|-
|February 27
|Mar. 13 <b>CANCELED</b>
|  
|Karin Melnick (University of Maryland)
|
|A D'Ambra Theorem in conformal Lorentzian geometry
|
|(Dymarz)
|-
|-
|March 6
|<b>Mar. 25</b> <b>CANCELED</b>
|[http://www3.nd.edu/~bwang3/ Botong Wang] (Notre Dame)
|Joerg Schuermann (University of Muenster, Germany)
|[[#Botong Wang (Notre Dame) |''Deformation theory with cohomology constraints.'']]
|An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles
|Max
|(Maxim)
|
|-
|-
|March 13
|Mar. 27 <b>CANCELED</b>
| [http://www.math.vanderbilt.edu/~saleaw/ Andrew Sale] (Vanderbilt)
|David Massey (Northeastern University)
|[[#Andrew Sale (Vanderbilt) | ''A geometric version of the conjugacy problem.'']]
|Extracting easily calculable algebraic data from the vanishing cycle complex
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|(Maxim)
|-
|March 20
|
|
|
|-
|March 27
|
|
|
|-
| Spring Break
|
|
|
|-
|April 10
|| [http://homepages.math.uic.edu/~mbhull/ Michael Hull] (UIC)
|[[#Michael Hull (UIC)|''Acylindrically hyperbolic groups'']]
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|-
| April 17
|| [https://sites.google.com/site/seanlimath/ Sean Li] (UChicago)
|[[#Sean Li (UChicago)|''Coarse differentiation of Lipschitz functions.'']]
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|-
|April 24
|
|
|
|-
|May 1
|| [http://www.math.sunysb.edu/~ssun/ Song Sun] (Stony Brook)
|[[#Song Sun (Stony Brook) | ''TBA'']]
|[http://www.math.wisc.edu/~bwang/ Wang]
|-
|
|
|-
|May 8
|
|
|
|-
|-
|<b>Apr. 10</b> <b>CANCELED</b>
|Antoine Song (Berkeley)
|TBA
|(Chen)
|}
|}


== Spring Abstracts ==
== Fall 2019 ==
 
===Balazs Strenner (Wisconsin)===
''Penner’s conjecture on pseudo-Anosov mapping classes.''
 
There are many constructions of pseudo-Anosov elements of mapping class groups of surfaces. Some of them are known to generate all pseudo-Anosov mapping classes, others are known not to. In 1988, Penner gave a very general construction of pseudo-Anosov mapping classes, and he conjectured that all pseudo-Anosov mapping classes arise this way up to finite power. This conjecture was known to be true on some simple surfaces, including the torus, but has otherwise remained open. In this talk I prove that the conjecture is false for most surfaces. (This  is joint work with Hyunshik Shin.)
 
===Ryan Budney (Victoria)===
''Operads and spaces of knots.''
 
I will describe a connection between the geometrization of 3-manifolds and a subject called operads.  It manifests itself as a structure theorem for the space of smooth embeddings of the circle in the 3-sphere and points to a homotopy-theoretic approach to old questions about finite-type invariants.
 
===Jenya  Sapir (UIUC)===
''Counting non-simple closed curves on surfaces.''
 
We show how to get coarse bounds on the number of (non-simple) closed geodesics on a surface, given upper bounds on both length and self-intersection number. Recent work by Mirzakhani has produced asymptotics for the growth of the number of simple closed curves, and this work has been extended to curves with at most 3 self-intersections with respect to length. However, no asymptotics, or even bounds, were previously known for other bounds on self-intersection number. Time permitting, we will discuss some applications of this result.
 
===Botong Wang (Notre Dame)===
''Deformation theory with cohomology constraints.''
 
Deformation theory is a powerful tool to study the local structure of moduli spaces.  I will first give an introduction to the theory of Deligne-Goldman-Millson, which translates deformation theory problems to problems of differential graded Lie algebras. I will also talk about a generalization to deformation theory problems with cohomology constraints. This is used to study the local structure of cohomology jump loci in various moduli spaces.
 
===Andrew Sale (Vanderbilt)===
''A geometric version of the conjugacy problem.''
 
The classic conjugacy problem of Max Dehn asks whether, for a given group, there is an algorithm that decides whether pairs of elements are conjugate. Related to this is the following question: given two conjugate elements u,v, what is the shortest length element w such that uw=wv? The conjugacy length function (CLF) formalises this question. I will survey what is known for CLFs of groups, giving a sketch proof for a result in semisimple Lie groups. I will also discuss a new, closely related function, the permutation conjugacy length function (PCL). I will outline its potential application to studying the computational complexity of the conjugacy problem, and describe a result, joint with Y. Antolin, for the PCL of relatively hyperbolic groups.
 
===Michael Hull (UIC)===
''Acylindrically hyperbolic groups''
 
Hyperbolic and relatively hyperbolic groups have
played an important role in the development of geometric group theory.
However, there are many other groups which admit interesting and useful
actions on hyperbolic metric spaces, including mapping class groups,
$Out(F_n)$, directly indecomposable RAAGs, and many 3-manifold groups.
The class of acylindrically hyperbolic groups provides a framework for
studying all of these groups (and many more) using many of the same
techniques developed for hyperbolic and relatively hyperbolic groups.
We will give a brief survey of examples and properties of
acylindrically hyperbolic groups and show how the study of this
class has yielded new results in a number of particular cases.
 
===Sean Li (UChicago)===
''Coarse differentiation of Lipschitz functions.''
 
Bates, Johnson, Lindenstrauss, Preiss, and Schechtman introduced a notion of large scale differentiation for Lipschitz functions between normed linear spaces.  We discuss an extension of this result to the nonabelian setting of Carnot groups and use it to derive quantitative estimates for nonembeddability of such groups into certain classes of metric spaces.
 
===Song Sun (Stony Brook)===
''TBA''
 
== Fall 2014==
 
 


{| cellpadding="8"
{| cellpadding="8"
Line 160: Line 68:
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|August 29
|Oct. 4
| Yuanqi Wang
|Ruobing Zhang (Stony Brook University)
| [[#Yuanqi Wang|''Liouville theorem for complex Monge-Ampere equations with conic singularities.'']]
| Geometric analysis of collapsing Calabi-Yau spaces
| [http://www.math.wisc.edu/~bwang Wang]
|(Chen)
|-
|September 5
|
|
|
|-
|September 12
| [http://people.uwec.edu/daviscw/ Chris Davis] (UW-Eau Claire)
| [[#Chris Davis (UW-Eau Claire)|''L<sup>2</sup> signatures and an example of Cochran-Harvey-Leidy'']]
| [http://www.math.wisc.edu/~maxim/ Maxim]
|-
|September 19
| [http://www.math.northwestern.edu/~knudsen/ Ben Knudsen] (Northwestern)
| [[#Ben Knudsen (Northwestern)|''Rational homology of configuration spaces via factorization homology'']]
| [http://www.math.wisc.edu/~ellenber/ Ellenberg]
|-
|September 26
|
|
|
|-
|October 3
| [http://homepages.math.uic.edu/~kwhyte/ Kevin Whyte] (UIC)
|[[#Kevin Whyte (UIC)|''Quasi-isometric embeddings of symmetric spaces'']]
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|-
|October 10
|[http://math.uchicago.edu/~akwalker/ Alden Walker] (UChicago)
|[[#Alden Walker (UChicago)|''Roots, Schottky Semigroups, and a proof of Bandt's Conjecture'']]
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|-
|October 17
|
|
|
|-
|October 24
|
|
|
|-
|October 31
| [http://www2.math.ou.edu/~jing/ Jing Tao] (Oklahoma)
| [[#Jing Tao (Oklahoma)|''Growth Tight Actions'']]
| [http://www.math.wisc.edu/~rkent/ Kent]
|-
|November 1
| [http://www.math.wisc.edu/~dymarz/yggt/ Young Geometric Group Theory in the Midwest Workshop]
|-
|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
| [http://www.math.upenn.edu/~alkju/ Alexandra Kjuchukova] (University of Pennsylvania)
| [[#Alexandra Kjuchukova (University of Pennsylvania)|''On the classification of irregular branched covers of four-manifolds'']]
| [http://www.math.wisc.edu/~Maxim/ Maxim]
|-
|-
|November 21
|Oct. 25
|  
|Emily Stark (Utah)
|
| Action rigidity for free products of hyperbolic manifold groups
|
|(Dymarz)
|-
|-
|Thanksgiving Recess
|Nov. 8
|  
|Max Forester (University of Oklahoma)
|
|Spectral gaps for stable commutator length in some cubulated groups
|
|(Dymarz)
|-
|-
|December 4, <b>Thursday at 4pm in VV 901</b>
|Nov. 22
| Oyku Yurttas (Georgia Tech)  
|Yu Li (Stony Brook University)
|[[#Oyku Yurttas (Georgia Tech)|''Dynnikov and train track transition matrices of pseudo-Anosov braids'']]
|On the structure of Ricci shrinkers
|[http://www.math.wisc.edu/~jeanluc/ Thiffeault]
|(Huang)
|-
|-
|December 5
| No seminar.
|
|
|-
|December 12
| No seminar.
|
|
|-
|
|}
|}


== Fall 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===
===Yuanqi Wang===
''Liouville theorem for complex Monge-Ampere equations with conic singularities.''
$G_{2}-$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.  


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


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


''Rational homology of configuration spaces via factorization homology''
===Joerg Schuermann===


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.
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.


===Kevin Whyte (UIC)===
===David Massey===
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)===
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.
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.
===Antoine Song===


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.
TBA


===Jing Tao (Oklahoma)===
==Fall Abstracts==
''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.
===Ruobing Zhang===


===Thomas Barthelm&eacute; (Penn State)===
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.
''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?
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.
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.
===Emily Stark===


===Alexandra Kjuchukova (University of Pennsylvania)===
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.
''On the classification of irregular branched covers of four-manifolds''


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


===Oyku Yurttas (Georgia Tech)===
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.
''Dynnikov and train track transition matrices of pseudo-Anosov braids''


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