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 2013==
== Spring 2020 ==
 
 


{| cellpadding="8"
{| cellpadding="8"
Line 16: Line 14:
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|September 6
|Feb. 7
|  
|Xiangdong Xie  (Bowling Green University)
|
| Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|
|(Dymarz)
|-
|-
|September 13, <b>10:00 AM in 901!</b>
|Feb. 14
| [http://www.ma.utexas.edu/users/zupan/ Alex Zupan] (Texas)
|Xiangdong Xie  (Bowling Green University)
| [[#Alex Zupan (Texas)| ''Totally geodesic subgraphs of the pants graph'']]
| Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
| [http://www.math.wisc.edu/~rkent/ Kent]
|(Dymarz)
|-
|-
|September 20
|Feb. 21
|  
|Xiangdong Xie  (Bowling Green University)
|
| Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|
|(Dymarz)
|-
|-
|September 27
|Feb. 28
|  
|Kuang-Ru Wu (Purdue University)
|
|Griffiths extremality, interpolation of norms, and Kahler quantization
|
|(Huang)
|-
|-
|October 4
|Mar. 6
|  
|Yuanqi Wang (University of Kansas)
|
|Moduli space of G2−instantons on 7−dimensional product manifolds
|
|(Huang)
|-
|-
|October 11
|Mar. 13 <b>CANCELED</b>
|  
|Karin Melnick (University of Maryland)
|
|A D'Ambra Theorem in conformal Lorentzian geometry
|
|(Dymarz)
|-
|-
|October 18
|<b>Mar. 25</b> <b>CANCELED</b>
| [http://www.math.uiuc.edu/~jathreya/ Jayadev Athreya] (Illinois)
|Joerg Schuermann (University of Muenster, Germany)
|[[#Jayadev Athreya (Illinois)| ''Gap Distributions and Homogeneous Dynamics'']]
|An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles
| [http://www.math.wisc.edu/~rkent/ Kent]
|(Maxim)
|-
|-
|October 25
|Mar. 27 <b>CANCELED</b>
| [http://www.math.wisc.edu/~robbin/ Joel Robbin (Wisconsin)]
|David Massey (Northeastern University)
| [[#Joel Robbin (Wisconsin) | ''GIT and <math>\mu</math>-GIT'']]
|Extracting easily calculable algebraic data from the vanishing cycle complex
| local
|(Maxim)
|-
|-
|November 1
|<b>Apr. 10</b> <b>CANCELED</b>
| [http://lukyanenko.net/ Anton Lukyanenko (Illinois)]
|Antoine Song (Berkeley)
| [[#Anton Lukyanenko (Illinois)| ''Uniformly quasi-regular mappings on sub-Riemannian manifolds'']]
|TBA
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|(Chen)
|}


== Fall 2019 ==
{| cellpadding="8"
!align="left" | date
!align="left" | speaker
!align="left" | title
!align="left" | host(s)
|-
|-
|November 8
|Oct. 4
| Neil Hoffman (Melbourne)
|Ruobing Zhang (Stony Brook University)
| [[#Neil Hoffman (Melbourne)| ''Verified computations for hyperbolic 3-manifolds'']]
| Geometric analysis of collapsing Calabi-Yau spaces
|[http://www.math.wisc.edu/~rkent/ Kent]
|(Chen)
|-
|November 15
| Khalid Bou-Rabee (Minnesota)
| [[#Khalid Bou-Rabee (Minnesota)| ''On generalizing a theorem of A. Borel'']]
|[http://www.math.wisc.edu/~rkent/ Kent]
|-
|-
|November 22
| Morris Hirsch (Wisconsin)
| [[#Morris Hirsch (Wisconsin)| ''Common zeros for Lie algebras of  vector fields on real and complex
2-manifolds.'']]
| local
|-
|-
|Thanksgiving Recess
|Oct. 25
|  
|Emily Stark (Utah)
|
| Action rigidity for free products of hyperbolic manifold groups
|
|(Dymarz)
|-
|-
|December 6
|Nov. 8
| Sean Paul (Wisconsin)
|Max Forester (University of Oklahoma)
| [[#Sean Paul (Wisconsin)| ''(Semi)stable Pairs I'']]
|Spectral gaps for stable commutator length in some cubulated groups
| local
|(Dymarz)
|-
|-
|December 13
|Nov. 22
| Sean Paul (Wisconsin)
|Yu Li (Stony Brook University)
| [[#Sean Paul (Wisconsin)| ''(Semi)stable Pairs II'']]
|On the structure of Ricci shrinkers
| local
|(Huang)
|-
|-
|
|}
|}


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


===Alex Zupan (Texas)===
===Xiangdong Xie===
''Totally geodesic subgraphs of the pants graph''


Abstract:
The quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces has played
For a compact surface S, the associated pants graph P(S) consists of vertices corresponding to pants decompositions of S and edges corresponding to elementary moves between pants decompositions.  Motivated by the Weil-Petersson geometry of Teichmüller space, Aramayona, Parlier, and Shackleton conjecture that the full subgraph G of P(S) determined by fixing a multicurve is totally geodesic in P(S).  We resolve this conjecture in the case that G is a product of Farey graphs. This is joint work with Sam Taylor.
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.


===Jayadev Athreya (Illinois)===
===Kuang-Ru Wu===
''Gap Distributions and Homogeneous Dynamics''
 
Abstract:
We discuss the notion of gap distributions of various lists of numbers in [0, 1], in particular focusing on those which are associated to certain low-dimensional dynamical systems. We show how to explicitly compute some examples using techniques of homogeneous dynamics, generalizing earlier work on gaps between Farey Fractions. This works gives some possible notions of `randomness' of special trajectories of billiards in polygons, and is based partly on joint works with J. Chaika, J. Chaika and S. Lelievre, and with Y.Cheung. This talk may also be of interest to number theorists.
 
===Joel Robbin (Wisconsin)===
GIT and  <math>\mu</math>-GIT
 
Many problems in differential geometry can be reduced to solving a PDE of form
<br><br>
<math>
    \mu(x)=0
</math>
<br><br>
where <math>x</math> ranges over some function space and <math>\mu</math> is an infinite dimensional analog of the moment map in symplectic geometry. 
In Hamiltonian dynamics the moment map was introduced to use a group action to reduce the number of degrees of freedom in the ODE.
It was soon discovered that the moment map could be applied to Geometric Invariant Theory:
if a  compact Lie group <math>G</math> acts on a projective algebraic variety <math>X</math>,
then the complexification <math>G^c</math> also acts and there is an isomorphism of orbifolds
<br><br>
<math>
    X^s/G^c=X//G:=\mu^{-1}(0)/G
</math>
<br><br>
between the space of orbits of Mumford's stable points and the Marsden-Weinstein quotient.


In September of 2013 Dietmar Salamon, his student Valentina Georgoulas, and I wrote an exposition of (finite dimensional) GIT from the point of view of symplectic geometry.  
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.
The theory works for compact Kaehler manifolds, not just projective varieties.
I will describe our paper in this talk; the following Monday Dietmar will give more details in the Geometric Analysis Seminar.


===Anton Lukyanenko (Illinois)===
===Yuanqi Wang===
''Uniformly quasi-regular mappings on sub-Riemannian manifolds''
$G_{2}-$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.


Abstract:
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.
A quasi-regular (QR) mapping between metric manifolds is a branched cover with bounded dilatation, e.g. f(z)=z^2. In a joint work with K. Fassler and K. Peltonen, we define QR mappings of sub-Riemannian manifolds and show that:
1) Every lens space admits a uniformly QR (UQR) mapping f.
2) Every UQR mapping leaves invariant a measurable conformal structure.
The first result uses an explicit "conformal trap" construction, while the second builds on similar results by Sullivan-Tukia and a connection to higher-rank symmetric spaces.


===Neil Hoffman (Melbourne)===
===Karin Melnick===
''Verified computations for hyperbolic 3-manifolds''


Abstract:
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.
Given a triangulated 3-manifold M a natural question is: Does M admit a hyperbolic structure?


While this question can be answered in the negative if M is known to
===Joerg Schuermann===
be reducible or toroidal, it is often difficult to establish a
certificate of hyperbolicity, and so computer methods have developed
for this purpose. In this talk, I will describe a new method to
establish such a certificate via verified computation and compare the
method to existing techniques.


This is joint work with Kazuhiro Ichihara, Masahide Kashiwagi,
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.
Hidetoshi Masai, Shin'ichi Oishi, and Akitoshi Takayasu.


===Khalid Bou-Rabee (Minnesota)===
===David Massey===
''On generalizing a theorem of A. Borel''


The proof of the Hausdorff-Banach-Tarski paradox relies on the existence of a nonabelian free group in the group of rotations of <math>\mathbb{R}^3</math>. To help generalize this paradox, Borel proved the following result on free groups.
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.


Borel’s Theorem (1983): Let <math>F</math> be a free group of rank two. Let <math>G</math> be an arbitrary connected semisimple linear algebraic group (i.e., <math>G = \mathrm{SL}_n</math> where <math>n \geq 2</math>). If <math>\gamma</math> is any  nontrivial element in <math>F</math> and <math>V</math> is any proper subvariety of <math>G(\mathbb{C})</math>, then there exists a homomorphism <math>\phi: F \to G(\mathbb{C})</math> such that <math>\phi(\gamma) \notin V</math>.
===Antoine Song===


What is the class, <math>\mathcal{L}</math>, of groups that may play the role of <math>F</math> in Borel’s Theorem? Since the free group of rank two is in <math>\mathcal{L}</math>, it follows that all residually free groups are in <math>\mathcal{L}</math>. In this talk, we present some methods for determining whether a finitely generated group is in <math>\mathcal{L}</math>. Using these methods, we give a concrete example of a finitely generated group in <math>\mathcal{L}</math> that is *not* residually free. After working out a few other examples, we end with a discussion on how this new theory provides an answer to a question of Brueillard, Green, Guralnick, and Tao concerning double word maps. This talk covers joint work with Michael Larsen.
TBA


===Morris Hirsch (Wisconsin)===
==Fall Abstracts==
''Common zeros for Lie algebras of  vector fields on real and complex 2-manifolds.''


The celebrated Poincare-Hopf theorem states that a vector field <math>X</math> on a manifold
===Ruobing Zhang===
<math>M</math> has nonempty zero set <math>Z(X)</math>, provided <math>M</math> is compact with empty boundary and
<math>M</math> has nonzero Euler characteristic. Surprising little is known about the set of
common zeros of two or more vector fields, especially when <math>M</math> is not compact.
One of the few results in this direction is a remarkable theorem of Christian
Bonatti (Bol. Soc. Brasil. Mat. 22 (1992), 215–247), stated below. When <math>Z(X)</math> is
compact, <math>i(X)</math> denotes the intersection number of <math>X</math> with the zero section of the
tangent bundle.


<math>\cdot </math> Assume <math> dim_{\mathbb{R}(M)} ≤ 4</math>, <math>X</math> is analytic, <math>Z(X)</math> is compact and <math>i(X) \neq 0</math>. Then
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.
every analytic vector field commuting with <math>X</math> has a zero in <math>Z(X)</math>.
In this talk I will discuss the following analog of Bonatti’s theorem. Let <math>\mathfrak{g}</math> be
a Lie algebra of analytic vector fields on a real or complex 2-manifold <math>M</math>, and set
<math>Z(g) := \cap_{Y \in \mathfrak{g}} Z(Y)</math>.


• Assume <math>X</math> is analytic, <math>Z(X)</math> is compact and <math>i(X) \neq 0</math>. Let <math>\mathfrak{g}</math> be generated by
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.
analytic vector fields <math>Y</math> on <math>M</math> such that the vectors <math>[X,Y]p</math> and <math>Xp</math> are linearly
dependent at all <math>p \in M</math>. Then <math>Z(\mathfrak{g}) \cap Z(X) \neq \emptyset </math>.
Related results on Lie group actions, and nonanalytic vector fields, will also be
treated.


===Sean Paul (Wisconsin)===
===Emily Stark===
''(Semi)stable Pairs I''


===Sean Paul (Wisconsin)===
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.
''(Semi)stable Pairs II''


===Max Forester===


== Spring 2014 ==
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===
{| cellpadding="8"
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.
!align="left" | date
!align="left" | speaker
!align="left" | title
!align="left" | host(s)
|-
|January 24
|
|
|
|-
|January 31
|[http://www.math.uiuc.edu/~dowdall/ Spencer Dowdall (UIUC)]
|[[#Spencer Dowdall (UIUC)| ''Fibrations and polynomial invariants for free-by-cyclic groups'']]
|[http://www.math.wisc.edu/~rkent Kent]
|
|-
|February 7
| Reserved for special Colloquium
|
|
|-
|February 14
|
|
|
|-
|February 21
| [http://www.math.vanderbilt.edu/~suvaini/ Ioana Suvaina (Vanderbilt)]
| [[#Ioana Suvaina (Vanderbilt)| ''TBA'']]
| [http://www.math.wisc.edu/~maxim/ Maxim]
|
|-
|February 28
|[http://gt.postech.ac.kr/~jccha/ Jae Choon Cha (POSTECH, Korea)]
|[[#Jae Choon Cha (POSTECH)| ''TBA'']]
|[http://www.math.wisc.edu/~maxim Maxim]
|
|-
|March 7
|
|
|
|-
|March 14
|
|
|
|-
|Spring Break
|
|
|
|-
|March 28
|
|
|
|-
| April 4
| [http://matthewkahle.org/ Matthew Kahle (Ohio)]
| [[#Matthew Kahle (Ohio)| ''TBA'']]
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|-
|April 11
|
|
|
|-
| April 18
| [https://www.math.lsu.edu/~pdani/ Pallavi Dani (LSU)]
| [[#Pallavi Dani (LSU)| ''TBA'']]
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|-
|April 25
| [http://www.math.sunysb.edu/~jsun/ Jingzhou Sun(Stony Brook)]
| [[#Jingzhou Sun(Stony Brook)| ''TBA'']]
|[Wang]
|-
|May 2
|
|
|
|-
|May 9
|
|
|
|-
|}
 
== Spring Abstracts ==
 
===Spencer Dowdall (UIUC)===
''Fibrations and polynomial invariants for free-by-cyclic groups''
 
The beautiful theory developed by Thurston, Fried and McMullen provides a near complete picture of the various ways a hyperbolic 3-manifold M can fiber over the circle. Namely, there are distinguished convex cones in the first cohomology M^1(M;R) whose integral points all correspond to fibrations of M, and the dynamical features of these fibrations are all encoded by McMullen's "Teichmuller polynomial."
 
This talk will describe recent work developing aspects of this picture in the setting of a free-by-cyclic group G. Specifically, I will introduce a polynomial invariant that determines a convex polygonal cone C in the first cohomology of G whose integral points all correspond to algebraically and dynamically interesting splittings of G. The polynomial invariant additionally provides a wealth of dynamical information about these splittings. This is joint work with Ilya Kapovich and Christopher J. Leininger.
 
===Matthew Kahle (Ohio)===
''TBA''
 
===Pallavi Dani (LSU)===
''TBA''
 
===Jingzhou Sun(Stony Brook)===
"TBA"


== 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]]
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]
<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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