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/~dymarz Tullia Dymarz] or [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova].
For more information, contact Shaosai Huang.


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


<!-- == Summer 2015 ==


== Spring 2020 ==


{| cellpadding="8"
{| cellpadding="8"
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!align="left" | host(s)
!align="left" | host(s)
|-
|-
|<b>June 23 at 2pm in Van Vleck 901</b>
|Feb. 7
| [http://www2.warwick.ac.uk/fac/sci/maths/people/staff/david_epstein/ David Epstein] (Warwick)
|Xiangdong Xie  (Bowling Green University)
| [[#David Epstein (Warwick) |''Splines and manifolds.'']]
| Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
| Hirsch
|(Dymarz)
|-
|-
|}
|Feb. 14
 
|Xiangdong Xie (Bowling Green University)
== Summer Abstracts ==
| Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
 
|(Dymarz)
===David Epstein (Warwick)===
''Splines and manifolds.''
 
[http://www.math.wisc.edu/~rkent/Abstract.Epstein.2015.pdf Abstract (pdf)]
 
-->
 
 
<!-- Spring 2016: [[Geometry_and_Topology_Seminar_Spring_2016]]
<br><br> -->
 
== Spring 2016 ==
 
In Spring of 2016, the seminar will meet on Thursdays at 2:25PM (location TBA).  
 
{| cellpadding="8"
!align="left" | date
!align="left" | speaker
!align="left" | title
!align="left" | host(s)
|-
|January 21
|
|
|
|-
|January 28
| [http://web.csulb.edu/~rblair/ Ryan Blair] (CSULB)
| [[#Ryan Blair|''Distance and Exceptional Surgeries on Knots'']]
| [http://www.math.wisc.edu/~kjuchukova Kjuchukova]
|-
|February 4
|   
| 
| 
|-
|February 11
|
|
|
|-
|February 18
|
|
|
|-
|February 25
|
|
|
|-
|March 3
| [https://math.la.asu.edu/~kotschwar/ Brett Kotschwar] (Arizona State University)
| [[#Brett Kotschwar|''TBA'']]
| [https://sites.google.com/a/wisc.edu/lu-wang/ L. Wang]
|-
|March 10
| [https://guests.mpim-bonn.mpg.de/pcahn/ Patricia Cahn] (Max Planck)
| [[#Patricia Cahn|''TBA'']]
| [http://www.math.wisc.edu/~kjuchukova Kjuchukova]
|-
|March 17
|
|
|
|-
|March 24
| Spring Break
|
|
|-
|-
|March 31
|Feb. 21
| [http://www.math.mcgill.ca/node/7171 Jingyin Huang] (McGill University)
|Xiangdong Xie  (Bowling Green University)
| [[#Jingyin Huang|"TBA"]]
| Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
| [http://www.math.wisc.edu/~dymarz Dymarz]
|(Dymarz)
|-
|-
|April 7
|Feb. 28
|  
|Kuang-Ru Wu (Purdue University)
|  
|Griffiths extremality, interpolation of norms, and Kahler quantization
|  
|(Huang)
|-
|-
|April 14
|Mar. 6
|  
|Yuanqi Wang (University of Kansas)
|  
|Moduli space of G2−instantons on 7−dimensional product manifolds
|  
|(Huang)
|-
|-
|April 21
|Mar. 13 <b>CANCELED</b>
| [http://personal.bgsu.edu/~xiex/ Xiangdong Xie] (Bowling Green University)
|Karin Melnick (University of Maryland)
| [[#Xiangdong Xie|"TBA"]]
|A D'Ambra Theorem in conformal Lorentzian geometry
| [http://www.math.wisc.edu/~dymarz Dymarz]
|(Dymarz)
|-
|-
|April 28
|<b>Mar. 25</b> <b>CANCELED</b>
|  
|Joerg Schuermann (University of Muenster, Germany)
|  
|An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles
|  
|(Maxim)
|-
|-
|May 5
|Mar. 27 <b>CANCELED</b>
| Gary Kennedy (TBC)
|David Massey (Northeastern University)
| TBA
|Extracting easily calculable algebraic data from the vanishing cycle complex
| Gonzalez Villa
|(Maxim)
|-
|-
|
|<b>Apr. 10</b> <b>CANCELED</b>
|Antoine Song (Berkeley)
|TBA
|(Chen)
|}
|}


== Spring Abstracts ==
== Fall 2019 ==
 
===Ryan Blair===
 
''Distance and Exceptional Surgeries on Knots''
 
Distance is a measure of complexity for a bicompressible surface in a 3-manifold which is defined using the curve complex for the surface. Recently, distance has been used to better understand Dehn surgery on knots in 3-manifolds. In particular, I will present results which show that knots with high distance surfaces do not admit non-hyperbolic surgeries or cosmetic surgeries. Applications to the cabling conjecture and the Berge conjecture will also be discussed.
 
== Fall 2015==
 
 


{| cellpadding="8"
{| cellpadding="8"
Line 146: Line 68:
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|September 4
|Oct. 4
|  
|Ruobing Zhang (Stony Brook University)
|
| Geometric analysis of collapsing Calabi-Yau spaces
|
|(Chen)
|-
|September 11
| [https://uwm.edu/math/people/tran-hung-1/ Hung Tran] (UW Milwaukee)
| [[#Hung Tran|''Relative divergence, subgroup distortion, and geodesic divergence'']]
| [http://www.math.wisc.edu/~dymarz T. Dymarz]
|-
|September 18
| [http://www.math.wisc.edu/~dymarz Tullia Dymarz] (UW Madison)
| [[#Tullia Dymarz|''Non-rectifiable Delone sets in amenable groups'']]
| (local)
|-
|September 25
| [https://jpwolfson.wordpress.com/ Jesse Wolfson] (Uchicago)
| [[#Jesse Wolfson|''Counting Problems and Homological Stability'']]
| [http://www.math.wisc.edu/~mmwood/ M. Matchett Wood]
|-
|October 2
| [https://riemann.unizar.es/~jicogo/ Jose Ignacio Cogolludo Agustín] (University of Zaragoza, Spain)
| [[#Jose Ignacio Cogolludo Agustín|''Topology of curve complements and combinatorial aspects'']]
|[http://www.math.wisc.edu/~maxim L. Maxim]
|-
|October 9
| [http://people.brandeis.edu/~mcordes/ Matthew Cordes] (Brandeis)
| [[#Matthew Cordes|''Morse boundaries of geodesic metric spaces'']]
| [http://www.math.wisc.edu/~dymarz T. Dymarz]
|-
|October 16
| [http://www.math.jhu.edu/~bernstein/ Jacob Bernstein] (Johns Hopkins University)
| [[#Jacob Bernstein (Johns Hopkins University)|''Hypersurfaces of low entropy'']]
| [http://www.sites.google.com/a/wisc.edu/lu-wang/ L. Wang]
|-
|October 23
| [https://sites.google.com/a/wisc.edu/ysu/ Yun Su] (UW Madison)
| [[#Yun Su (Brandeis)|''Higher-order degrees of hypersurface complements.'']]
| (local)
|-
|October 30
| [http://www.math.stonybrook.edu/phd-student-directory Gao Chen] (Stony Brook University)
| [[#Gao Chen(Stony Brook University)|''Classification of gravitational instantons '']]
| [http://www.math.wisc.edu/~bwang B.Wang]
|-
|November 6
| [http://scholar.harvard.edu/gardiner Dan Cristofaro-Gardiner] (Harvard)
| [[#Dan Cristofaro-Gardiner|''Higher-dimensional symplectic embeddings and the Fibonacci staircase'']]
| [http://www.math.wisc.edu/~kjuchukova Kjuchukova]
|
|-
|-
|November 13
| [http://people.brandeis.edu/~ruberman/ Danny Ruberman] (Brandeis)
| [[#Danny Ruberman|''Configurations of embedded spheres'']]
| [http://www.math.wisc.edu/~kjuchukova Kjuchukova]
|
|-
|-
|November 20
|Oct. 25
| [https://www.math.toronto.edu/cms/izosimov-anton/ Anton Izosimov] (University of Toronto)
|Emily Stark (Utah)
| [[#Anton Izosimov (University of Toronto)|''Stability for the multidimensional rigid body and singular curves'']]
| Action rigidity for free products of hyperbolic manifold groups
| [http://www.math.wisc.edu/~maribeff/ Mari-Beffa]
|(Dymarz)
|-
|-
|Thanksgiving Recess
|Nov. 8
|  
|Max Forester (University of Oklahoma)
|
|Spectral gaps for stable commutator length in some cubulated groups
|
|(Dymarz)
|-
|-
|December 4
|Nov. 22
| [http://www.math.wisc.edu/~westrich/ Quinton Westrich] (UW Madison)
|Yu Li (Stony Brook University)
| [[#Quinton Westrich|''Harmonic Chern Forms on Polarized Kähler Manifolds'']]
|On the structure of Ricci shrinkers
| (local)
|(Huang)
|-
|-
|December 11
|[http://kaihowong.weebly.com/ Tommy Wong] (UW Madison)
| [[#Tommy Wong|''Milnor Fiber of Complex Hyperplane Arrangement'']]
| (local)
|-
|
|}
|}


== Fall Abstracts ==
==Spring Abstracts==
 
 
===Hung Tran===
''Relative divergence, subgroup distortion, and geodesic divergence''
 
In my presentation, I introduce three new invariants for pairs $(G;H)$ consisting of a finitely generated group $G$ and a subgroup $H$. The first invariant is the upper relative divergence which generalizes Gersten's notion of divergence. The second invariant is the lower relative divergence which generalizes a definition of Cooper-Mihalik. The third invariant is the lower subgroup distortion which parallels the standard notion
of subgroup distortion. We examine the relative divergence (both upper and lower) of a group with respect to a normal subgroup or a cyclic subgroup. We also explore relative divergence of $CAT(0)$ groups and relatively hyperbolic groups with respect to various subgroups to better understand geometric properties of these groups. We answer the question of Behrstock and Drutu about the existence of Morse geodesics in $CAT(0)$ spaces with divergence function strictly greater than $r^n$ and strictly less than $r^{n+1}$, where $n$ is an integer greater than $1$. More precisely, we show that for each real number $s>2$, there is a $CAT(0)$ space $X$ with a proper and cocompact action of some finitely generated group such that $X$ contains a Morse bi-infinite geodesic with the divergence equivalent to $r^s$.
 
 
===Tullia Dymarz===
''Non-rectifiable Delone sets in amenable groups''


In 1998 Burago-Kleiner and McMullen constructed the first
===Xiangdong Xie===
examples of coarsely dense and uniformly discrete subsets of R^n that are
not biLipschitz equivalent to the standard lattice Z^n. Similarly we
find subsets inside the three dimensional solvable Lie group SOL that are
not bilipschitz to any lattice in SOL. The techniques involve combining
ideas from Burago-Kleiner with quasi-isometric rigidity results from
geometric group theory.


===Jesse Wolfson===
The quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces has played
''Counting Problems and Homological Stability''
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.


In 1969, Arnold showed that the i^{th} homology of the space of un-ordered configurations of n points in the plane becomes independent of n for n>>i. A decade later, Segal extended Arnold's method to show that the i^{th} homology of the space of degree n holomorphic maps from \mathbb{P}^1 to itself also becomes independent of n for large n, and, moreover, that both sequences of spaces have the same limiting homology. We explain how, using Weil's number field/function field dictionary, one might have predicted this topological coincidence from easily verifiable statements about specific counting problems.  We then discuss ongoing joint work with Benson Farb and Melanie Wood in which we use other counting problems to predict and discover new instances of homological stability in the topology of complex manifolds.
===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.


===Matthew Cordes===
===Yuanqi Wang===
''Morse boundaries of geodesic metric spaces''
$G_{2}-$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.


I will introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with equivalence classes of geodesic rays that identify the ``hyperbolic directions" in that space. (A ray is Morse if quasi-geodesics with endpoints on the ray stay bounded distance from the ray.) This boundary is a quasi-isometry invariant and a visibility space. In the case of a proper CAT(0) space the Morse boundary generalizes the contracting boundary of Charney and Sultan and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. Time permitting I will also discuss some results on Morse boundary of the mapping class group and briefly describe joint work with David Hume developing a capacity dimension for the Morse boundary.
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.


===Anton Izosimov===
===Karin Melnick===
''Stability for the multidimensional rigid body and singular curves''


A classical result of Euler says that the rotation of a
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.
torque-free 3-dimensional rigid body about the short or the long axis is
stable, while the rotation about the middle axis is unstable. I will
present a multidimensional generalization of this result and explain how
it can be proved using some basic algebraic geometry of singular curves.


===Jacob Bernstein===
===Joerg Schuermann===
''Hypersurfaces of low entropy''


The entropy is a quantity introduced by Colding and Minicozzi and may be thought of as a rough measure of the geometric complexity of a hypersurface of Euclidean space.  It is closely related to the mean curvature flow.  On the one hand, the entropy controls the dynamics of the flow. On the other hand, the mean curvature flow may be used to study the entropy.  In this talk I will survey some recent results with Lu Wang that show that hypersurfaces of low entropy really are simple.
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.


===Yun Su===
===David Massey===
''Higher-order degrees of hypersurface complements.''


===Gao Chen===
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.
''Classification of gravitational instantons''


A gravitational instanton is a noncompact complete hyperkahler manifold of real dimension 4 with faster than quadratic curvature decay. In this talk, I will discuss the recent work towards the classification of gravitational instantons. This is a joint work with X. X. Chen.
===Antoine Song===


===Dan Cristofaro-Gardiner===
TBA
''Higher-dimensional symplectic embeddings and the Fibonacci staircase''


McDuff and Schlenk determined when a four dimensional symplectic ellipsoid can be embedded into a ball, and found that when the ellipsoid is close to round, the answer is given by an infinite staircase determined by the odd-index Fibonacci numbers.  I will explain joint work with Richard Hind, showing that a generalization of this holds in all even dimensions.
==Fall Abstracts==


===Danny Ruberman===
===Ruobing Zhang===
''Configurations of embedded spheres''


Configurations of lines in the plane have been studied since antiquity. In recent years, combinatorial methods have been used to decide if a specified incidence relation between certain objects ("lines") and other objects ("points") can be realized by actual points and lines in a projective plane over a field. For the real and complex fields, one can weaken the condition to look for topologically embedded lines (circles in the real case, spheres in the complex case) that meet according to a specified incidence relation. I will explain some joint work with Laura Starkston (Stanford) giving new topological restrictions on the realization of configurations of spheres in the complex projective plane.
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.


===Quinton Westerich===
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.


''Harmonic Chern Forms on Polarized Kähler Manifolds''
===Emily Stark===


Abstract: The higher K-energies are functionals whose critical points 
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.
give Kähler metrics with harmonic Chern forms. In this talk, we relate 
the higher K-energies to discriminants and use the theory of stable 
pairs to obtain results on their boundedness and asymptotics.


===Max Forester===


===Tommy Wong===
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.


''Milnor Fiber of Complex Hyperplane Arrangement''
===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.
The existence of Milnor fibration creates rooms and provides a platform to discuss the topology of complex algebraic varieties. In this talk, the study of hyperplane arrangements will be specified.
Many open questions have been raised subject to the Milnor fiber of the mentioned fibration. For instance,while the homology of the arrangement complement can be described by the Orlik-Soloman Algebra, which is combinatorically determined by the intersection poset, it has been conjectured that the poset also determines the homology of the Milnor fiber.
There are active work on this open conjecture, especially in C^3. Several classical results will be mentioned in the talk. A joint work with Su, serving as an improvement of some of the classical work, will also be briefly described.


== 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]]
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]
<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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