Geometry and Topology Seminar 2019-2020: Difference between revisions

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Traditionally, 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> In Spring of 2016, the seminar will meet on '''Thursdays''' at 2:25PM in B 231 of Van Vleck. <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 ==


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|-
|<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 in B 231 of Van Vleck.
 
{| 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
|Feb. 21
|
|Xiangdong Xie  (Bowling Green University)
|
| Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|
|(Dymarz)
|-
|-
|February 11
|Feb. 28
| [http://www.math.wisc.edu/~wang Botong Wang] (UW Madison)
|Kuang-Ru Wu (Purdue University)
| [[#Botong Wang|''A family of Symplectic-Complex Calabi-Yau Manifolds that are NonKahler'']]
|Griffiths extremality, interpolation of norms, and Kahler quantization
| (local)
|(Huang)
|-
|-
|February 18
|Mar. 6
|  
|Yuanqi Wang (University of Kansas)
|  
|Moduli space of G2−instantons on 7−dimensional product manifolds
|  
|(Huang)
|-
|-
|February 25
|Mar. 13 <b>CANCELED</b>
| [https://www.math.purdue.edu/~psolapur/ Partha Solapurkar] (Purdue University)
|Karin Melnick (University of Maryland)
| [[#Partha Solapurkar|''Some new surfaces of general type with maximal Picard number'']]
|A D'Ambra Theorem in conformal Lorentzian geometry
| [http://www.math.wisc.edu/~wang Botong Wang]
|(Dymarz)
|-
|-
|March 3
|<b>Mar. 25</b> <b>CANCELED</b>
| [https://math.la.asu.edu/~kotschwar/ Brett Kotschwar] (Arizona State University)
|Joerg Schuermann (University of Muenster, Germany)
| [[#Brett Kotschwar|''Ricci flow and bounded curvature'']]
|An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles
| [https://sites.google.com/a/wisc.edu/lu-wang/ L. Wang]
|(Maxim)
|-
|-
|Friday March 4 (1:20-2:10pm in VV B211)
|Mar. 27 <b>CANCELED</b>
| [http://homepages.math.uic.edu/~cantrell/ Mike Cantrell] (UIC)
|David Massey (Northeastern University)
| [[#Mike Cantrell|'"Asymptotic shapes for ergodic families of metrics on nilpotent groups"]]
|Extracting easily calculable algebraic data from the vanishing cycle complex
| [http://www.math.wisc.edu/~dymarz Dymarz]
|(Maxim)
|-
|-
|Wednesday March 9 (1:20-2:10pm in VV B211)
|<b>Apr. 10</b> <b>CANCELED</b>
| [http://www.uncg.edu/~t_fernos/ Talia Fernos] (Greensboro)
|Antoine Song (Berkeley)
| [[#Talia Fernos|''The Roller Compactification and CAT(0) Cube Complexes'']]
|TBA
| [http://www.math.wisc.edu/~dymarz Dymarz]
|(Chen)
|-
|March 10
| [https://guests.mpim-bonn.mpg.de/pcahn/ Patricia Cahn] (Max Planck)
| [[#Patricia Cahn|''Knots Transverse to a Vector Field'']]
| [http://www.math.wisc.edu/~kjuchukova Kjuchukova]
|-
|March 17
|
|
|
|-
|March 24
| Spring Break
|
|
|-
|March 31
| [http://www.math.mcgill.ca/node/7171 Jingyin Huang] (McGill University)
| [[#Jingyin Huang|"TBA"]]
| [http://www.math.wisc.edu/~dymarz Dymarz]
|-
|April 7
|
|
|
|-
|April 14
|
|
|
|-
|April 21
| [http://personal.bgsu.edu/~xiex/ Xiangdong Xie] (Bowling Green University)
| [[#Xiangdong Xie|"TBA"]]
| [http://www.math.wisc.edu/~dymarz Dymarz]
|-
|April 28
| [http://cgp.ibs.re.kr/~yongoh/ Yong-Geun Oh] (IBS Center for Geometry and Physics & Postech)  
| [[#Yong-Geun Oh| "TBA"]]
|
|-
|May 5
| [https://people.math.osu.edu/kennedy.28/ Gary Kennedy] (Ohio State University)
| TBA
| Gonzalez Villa
|-
|
|}
|}


== 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.
 
===Brett Kotschwar===
 
''Ricci flow and bounded curvature''
 
The problem of determining when a given solution to the Ricci flow with initially bounded curvature will continue to have bounded curvature has bearing on both the uniqueness and long-time existence of solutions to the flow. I will discuss two results in this direction which are equally valid in the noncompact setting, the first based on a simple proof of an extension to the standard uniqueness theorem of Hamilton and Chen-Zhu, and the second, joint with Ovidiu Munteanu and Jiaping Wang, based on new explicit local estimates of the curvature under a uniform bound on the Ricci tensor.
 
===Partha Solapurkar===
 
''Some new surfaces of general type with maximal Picard number''
 
The Picard number $ \rho(X) $ of a surface $ X $ is the rank of its Neron-Severi group. It is bounded above by the Hodge number $ h^{11}(X) $. We say that a surface has maximal Picard number if it has the largest possible Picard number: $ \rho(X) = h^{11}(X) $. In 1972, Shioda constructed elliptic modular surfaces and among other things, proved that they have maximal Picard number. Our first idea is to take elliptic modular surfaces and replace each elliptic curve with a canonically constructed genus 2 curve. Under nice circumstances the resulting surfaces do indeed have maximal Picard number. There is also a second set of examples that arise as the total space of the moduli space of quaternionic Shimura curves. This is joint work with my advisor Prof. D. Arapura.
 
===Botong Wang===
 
''A family of Symplectic-Complex Calabi-Yau Manifolds that are NonKahler''
 
A Kahler manifold is a smooth manifold with compatible complex and symplectic structures. In general, a compact manifold which admits both complex and symplectic structures may not admit any Kahler structure. Hodge theory and hard Lefschetz theorem have very strong implications on the homotopy type of compact Kahler manifolds. We introduce a family of 6-dimensional compact manifolds $M(A)$, which admit both Calabi-Yau symplectic and Calabi-Yau complex structures. They satisfy all the consequences of classical Hodge theory and hard Lefschetz theorem. However, we show that they are not homotopy equivalent to any compact Kahler manifold using a recently developed cohomology jump loci method. This is joint work with Lizhen Qin.
 
===Mike Cantrell===
"Asymptotic shapes for ergodic families of metrics on nilpotent groups"
 
Let G be a finitely generated virtually nilpotent group. We consider three closely related problems: (i) convergence to a deterministic asymptotic cone for an equivariant ergodic family of inner metrics on G, generalizing Pansu’s theorem; (ii) the asymptotic shape theorem for First Passage Percolation for general (not necessarily independent) ergodic processes on edges of a Cayley graph of G; (iii) the sub-additive ergodic theorem over a general ergodic G-action. The
limiting objects are given in terms of a Carnot-Carathérodory metric on the graded nilpotent group associated to the Mal’cev completion of G.
 
===Talia Fernos===
 
"The Roller Compactification and CAT(0) Cube Complexes"
 
The Roller compactification is a beautiful object that ties together the combinatorial and geometric properties that characterize CAT(0) cube complexes. In this talk, we will discuss this compactification in the context of superrigidity, the fixed point Property (FC), and the abundance of regular elements (i.e. automorphisms that are rank-1 in each irreducible factor). These results are collaborations with Caprace, Chatterji, and Iozzi, as well as, Lécureux, and Mathéus.
 
 
 
===Patricia Cahn===
 
"Knots Transverse to a Vector Field"
 
We study knots transverse to a fixed vector field V on a 3-manifold M up to the corresponding isotopy relation. Such knots are equipped with a natural framing.  Motivated by questions in contact topology, it is natural to ask whether two V-transverse knots which are isotopic as framed knots and homotopic through V-transverse immersed curves must be isotopic through V-transverse knots.  When M is R^3 and V is the vertical vector field the answer is yes.  However, we construct examples which show the answer to this question can be no in other 3-manifolds, specifically S^1-fibrations over surfaces of genus at least 2.  We also give a general classification of knots transverse to a vector field in an arbitrary closed oriented 3-manifold M.  We show this classification is particularly simple when V is the co-orienting vector field of a tight contact structure, or when M is irreducible and atoroidal. Lastly, we apply our results to study loose Legendrian knots in overtwisted contact manifolds, and generalize results of Dymara and Ding-Geiges.  This work is joint with Vladimir Chernov.
 
== Fall 2015==
 
 


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|-
|-
|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
|Oct. 25  
| [https://jpwolfson.wordpress.com/ Jesse Wolfson] (Uchicago)
|Emily Stark (Utah)
| [[#Jesse Wolfson|''Counting Problems and Homological Stability'']]
| Action rigidity for free products of hyperbolic manifold groups
| [http://www.math.wisc.edu/~mmwood/ M. Matchett Wood]
|(Dymarz)
|-
|-
|October 2
|Nov. 8
| [https://riemann.unizar.es/~jicogo/ Jose Ignacio Cogolludo Agustín] (University of Zaragoza, Spain)
|Max Forester (University of Oklahoma)
| [[#Jose Ignacio Cogolludo Agustín|''Topology of curve complements and combinatorial aspects'']]
|Spectral gaps for stable commutator length in some cubulated groups
|[http://www.math.wisc.edu/~maxim L. Maxim]
|(Dymarz)
|-
|-
|October 9
|Nov. 22
| [http://people.brandeis.edu/~mcordes/ Matthew Cordes] (Brandeis)
|Yu Li (Stony Brook University)
| [[#Matthew Cordes|''Morse boundaries of geodesic metric spaces'']]
|On the structure of Ricci shrinkers
| [http://www.math.wisc.edu/~dymarz T. Dymarz]
|(Huang)
|-
|-
|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
| [https://www.math.toronto.edu/cms/izosimov-anton/ Anton Izosimov] (University of Toronto)
| [[#Anton Izosimov (University of Toronto)|''Stability for the multidimensional rigid body and singular curves'']]
| [http://www.math.wisc.edu/~maribeff/ Mari-Beffa]
|-
|Thanksgiving Recess
|
|
|
|-
|December 4
| [http://www.math.wisc.edu/~westrich/ Quinton Westrich] (UW Madison)
| [[#Quinton Westrich|''Harmonic Chern Forms on Polarized Kähler Manifolds'']]
| (local)
|-
|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
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===
===Xiangdong Xie===
''Counting Problems and Homological Stability''


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.  
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===


===Matthew Cordes===
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.
''Morse boundaries of geodesic metric spaces''


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.
===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.  


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


A classical result of Euler says that the rotation of a
===Karin Melnick===
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===
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.
''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.
===Joerg Schuermann===


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


===Gao Chen===
===David Massey===
''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.
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.


===Dan Cristofaro-Gardiner===
===Antoine Song===
''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.
TBA


===Danny Ruberman===
==Fall Abstracts==
''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.
===Ruobing Zhang===


===Quinton Westerich===
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.


''Harmonic Chern Forms on Polarized Kähler Manifolds''
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.


Abstract: The higher K-energies are functionals whose critical points 
===Emily Stark===
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.


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.


===Tommy Wong===
===Max Forester===


''Milnor Fiber of Complex Hyperplane Arrangement''
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.


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.
===Yu Li===
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.
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.
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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