Geometry and Topology Seminar 2019-2020: Difference between revisions

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The [[Geometry and Topology]] seminar meets in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm.
The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.
<br>
<br>  
For more information, contact [http://www.math.wisc.edu/~rkent Richard Kent].
For more information, contact Shaosai Huang.


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


== Spring 2013 ==


== Spring 2020 ==


{| cellpadding="8"
{| cellpadding="8"
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!align="left" | host(s)
!align="left" | host(s)
|-
|-
|January 25
|Feb. 7
| [http://www.maths.usyd.edu.au/u/athomas/ Anne Thomas] (Sydney)
|Xiangdong Xie  (Bowling Green University)
| [[#Anne Thomas (Sydney)| ''Divergence in right-angled Coxeter groups'']]
| Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|(Dymarz)
|-
|-
|February 1
|Feb. 14
|
|Xiangdong Xie  (Bowling Green University)
|
| Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|
|(Dymarz)
|-
|-
|February 8
|Feb. 21
|
|Xiangdong Xie  (Bowling Green University)
|
| Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|
|(Dymarz)
|-
|-
|February 15
|Feb. 28
| [http://www3.nd.edu/~lnicolae/ Liviu Nicolaescu] (Notre Dame)
|Kuang-Ru Wu (Purdue University)
| [[#Liviu Nicolaescu (Notre Dame)| ''Random Morse functions and spectral geometry'']]
|Griffiths extremality, interpolation of norms, and Kahler quantization
|[http://www.math.wisc.edu/~oh/ Oh]
|(Huang)
|-
|-
|February 22
|Mar. 6
|
|Yuanqi Wang (University of Kansas)
|
|Moduli space of G2−instantons on 7−dimensional product manifolds
|
|(Huang)
|-
|-
|March 1
|Mar. 13 <b>CANCELED</b>
| [https://pantherfile.uwm.edu/chruska/www/ Chris Hruska] (UW Milwaukee)
|Karin Melnick (University of Maryland)
| [[#Chris Hruska (UW Milwaukee)| ''Local topology of boundaries and isolated flats'']]
|A D'Ambra Theorem in conformal Lorentzian geometry
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|(Dymarz)
|-
|-
|March 8
|<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)
|-
|-
|March 11, <b>MONDAY in B113!</b>
|Mar. 27 <b>CANCELED</b>
| [http://www.math.fsu.edu/~hironaka/ Eriko Hironaka] (FSU)
|David Massey (Northeastern University)
| [[#Eriko Hironaka (FSU)| ''Small dilatation pseudo-Anosov mapping classes'']]
|Extracting easily calculable algebraic data from the vanishing cycle complex
|[http://www.math.wisc.edu/~rkent/ Kent]
|(Maxim)
|-
|March 15
| Yu-Shen Lin (Harvard)
| [[#Yu-Shen Lin (Harvard)| ''Open Gromov-Witten Invariants on K3 surfaces and Wall-Crossing'']]
| [http://www.math.wisc.edu/~oh/ Oh]
|-
|March 20 <b>WEDNESDAY in 901!</b>
|[http://www.math.nyu.edu/faculty/cappell/index.html Sylvain Cappell] (NYU)
|[[#Sylvain Cappell (NYU)| ''Topological actions of compact, connected Lie Groups on Manifolds'']]
| [http://www.math.wisc.edu/~maxim/ Maxim]
|-
|Spring Break
|
|
|
|-
|April 5
|
|
|
|-
|April 12
|[http://www.mathi.uni-heidelberg.de/~villa/ Manuel Gonzalez Villa] (Heidelberg)
|[[#Manuel Gonzalez Villa (Heidelberg)| '' The monodromy conjecture for plane meromorphic germs'']]
|Maxim
|-
|April 19
|
|
|
|-
|April 26
| Emmy Murphy (MIT)
| [[#Emmy Murphy (MIT) | ''Exact Lagrangian immersions with few transverse self intersections'']]
| [http://www.math.wisc.edu/~oh/ Oh]
|-
|May 3
| Yuan-qi Wang (UCSB)
| [[#Yuan-qi Wang (UCSB)| ''Bessel Functions, Heat Kernel and the Conical Kahler-Ricci 
Flow'']]
| [http://www.math.wisc.edu/~bwang/ Wang]
|-
|
|
|-
|May 10
| [http://www.math.wisc.edu/~oh/ Yong-Geun Oh] (Wisconsin)
| [[#Yong-Geun Oh (Wisconsin)| ''Analysis of contact instantons and contact homology'']]
| Local
|-
|-
|<b>Apr. 10</b> <b>CANCELED</b>
|Antoine Song (Berkeley)
|TBA
|(Chen)
|}
|}


== Spring Abstracts ==
== Fall 2019 ==
 
===Anne Thomas (Sydney)===
''Divergence in right-angled Coxeter groups''
 
Abstract:
The divergence of a pair of geodesic rays emanating from a point is a
measure of how quickly they are moving away from each other. In
Euclidean space divergence is linear, while in hyperbolic space
divergence is exponential. Gersten used this idea to define a
quasi-isometry invariant for groups, also called divergence, which has
been investigated for classes of groups including fundamental groups
of 3-manifolds, mapping class groups and right-angled Artin groups. I
will discuss joint work with Pallavi Dani on divergence in
right-angled Coxeter groups (RACGs).  We characterise 2-dimensional
RACGs with quadratic divergence, and prove that for every positive
integer d, there is a RACG with divergence polynomial of degree d.
 
===Liviu Nicolaescu (Notre Dame)===
''Random Morse functions and spectral geometry''
 
Abstract:
I will discuss the distribution of critical values of a  smooth random function on a compact m-dimensional Riemann manifold (M,g)  described as a random  superposition  of eigenfunctions of the Laplacian.  The  notion of randomness that we use    has a naturally  built in  small parameter $\varepsilon$, and we show that    as $\varepsilon\to 0$ the distribution of critical  values closely resemble the distribution  of eigenvalues  of  certain  random symmetric  $(m+1)\times (m+1)$-matrices  of the type introduced by E. Wigner  in quantum mechanics. Additionally, I will  explain how to recover the metric  $g$ from  statistical  properties of the Hessians of the above random function.
 
===Chris Hruska (UW Milwaukee)===
''Local topology of boundaries and isolated flats''
 
Abstract:  Swarup proved that every one-ended word hyperbolic group has a
locally connected Gromov boundary.  However for CAT(0) groups,
non-locally connected boundaries are easy to construct.  For instance
the boundary of F_2 x Z is the suspension of a Cantor set.
 
In joint work with Kim Ruane, we have studied boundaries of CAT(0)
spaces with isolated flats.  If G acts properly, cocompactly on such a
space X, we give a necessary and sufficient condition on G such that
the boundary of X is locally connected.  As a corollary, we deduce
that such a group G is semistable at infinity.
 
===Eriko Hironaka (FSU)===
''Small dilatation pseudo-Anosov mapping classes''
 
The theory of fibered faces implies that  pseudo-Anosov
mapping classes with bounded normalized dilatation can be partitioned
into a finite number of families with related dynamics.  In this talk we
discuss the problem of finding concrete description
of the members of these families.  One conjectural way generalizes a
well-known sequence
defined by Penner in '91.  However, so far no known examples  of
this type come close to
the smallest known accumulation point of normalized dilatations.
In this talk we describe a different construction that uses mixed-sign
Coxeter systems.  A deformation of the simplest pseudo-Anosov braid monodromy
can be obtained in this way, and hence this model does realize the
smallest known accumulation point.
 
===Yu-Shen Lin (Harvard)===
''Open Gromov-Witten Invariants on K3 surfaces and Wall-Crossing''
 
Strominger-Yau-Zaslow conjecture suggests that the Ricci-flat metric on Calabi-Yau manifolds might be related to holomorphic discs. In this talk, I will define a new open Gromov-Witten invariants on elliptic K3 surfaces trying to explain this conjecture. The new invariant satisfies certain wall-crossing formula and multiple cover formula. I will also establish a tropical-holomorphic correspondence. Moreover, this invariant is expected to be equivalent to the generalized Donaldson-Thomas invariants in the hyperK\"ahler metric constructed by Gaiotto-Moore-Neitzke. If time allowed, I will talk about the connection with disks counting on Calabi-Yau 3-folds.
 
===Sylvain Cappell (NYU)===
''TBA''
 
===Manuel Gonzalez Villa (Heidelberg)===
''The monodromy conjecture for plane meromorphic germs''
 
Joint work with  Ann Lemahieu (Lille). A notion of Milnor fibration  for meromorphic functions and the corresponding concepts of  monodromy and monodromy zeta function, introduced by Gussein-Zade, Luengo and Melle, invite to consider the notion of  topological zeta function for meromorphic germs and the corresponding monodromy conjecture. We try to motive these notions and discuss the plane case. We show that the poles do not behave as in the holomorphic case but still do satisfy a generalization of the monodromy conjecture.
 
===Emmy Murphy (MIT)===
''Exact Lagrangian immersions with few transverse self intersections''
 
This talk will focus on the following question: supposing a
smooth manifold immerses into C^n as an exact Lagrangian, what is the
minimal number of transverse self-intersections necessary? Finding lower
bounds on the number of intersections of two embedded Lagrangians is a
central problem in symplectic topology which has seen much success; in
contrast bounding the number of self-intersections of an exact Lagrangian
immersion requires more advanced tools and the known results are far less
general. We show that no Arnold-type lower bound exists for exact
Lagrangian immersions by constructing examples with surprisingly few
self-intersections. For example, we show that any three-manifold immerses
as an exact Lagrangian in C^3 with a single transverse self-intersection.
We also apply Lagrangian surgery to these immersions to give some
interesting new examples of Lagrangian embeddings. (This is joint work of
the speaker with T. Ekholm, Y. Eliashberg, and I. Smith.)
 
===Yuan-qi Wang (UCSB)===
''Bessel Functions, Heat Kernel and the Conical Kahler-Ricci 
Flow''
 
Inspired by Donaldson's program, we introduce the Kahler 
Ricci flow with conical singularities. The main part of this talk is 
to show that the conical Kahler Ricci flow exists for short time and 
for long time in a proper space. These existence results are highly 
related to heat kernel and Bessel functions. We will also discuss some 
easy applications of the conical Kahler Ricci flow in conical Kahler 
geometry.
 
===Yong-Geun Oh (Wisconsin)===
''Analysis of contact instantons and contact homology''
 
In this talk, we explain the analysis of the following system of (degenerate) elliptic equation
$$
\overline \partial^\pi w = 0, \, d(w^*\lambda \circ j) = 0
$$
associated for each given contact triad $(Q,\lambda,J)$ on a contact manifold $(Q,\xi)$.
(Such an equation was first introduced by Hofer.)  We directly work with this equation
on the contact manifold without involving the symplectization process. We explain the basic
analytical ingredients towards the construction of moduli space of
solutions, which we call contact instantons.  I will indicate how one can define contact
homology type invariants using such a moduli space, which is still in progress. The talk
is partially based on the joint work with Rui Wang.
 
 
== Fall 2012==
 
 


{| cellpadding="8"
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!align="left" | host(s)
|-
|-
|September 21
|Oct. 4
| [http://www.math.wisc.edu/~josizemore/ Owen Sizemore] (Wisconsin)
|Ruobing Zhang (Stony Brook University)
| [[#Owen Sizemore (Wisconsin) |
| Geometric analysis of collapsing Calabi-Yau spaces
''Operator Algebra Techniques in Measureable Group Theory'']]
|(Chen)
| local
|-
|September 28
|[https://engineering.purdue.edu/~mboutin/ Mireille Boutin] (Purdue)
|[[#Mireille Boutin (Purdue) |
''The Pascal Triangle of a discrete Image: <br>
definition, properties, and application to object segmentation'']]
|[http://www.math.wisc.edu/~maribeff/ Mari Beffa]
|-
|October 5
| [http://www.math.msu.edu/~schmidt/ Ben Schmidt] (Michigan State)
| [[#Ben Schmidt (Michigan State)|
''Three manifolds of constant vector curvature'']]
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|-
|October 12
| [https://www2.bc.edu/ian-p-biringer/ Ian Biringer] (Boston College)
| [[#Ian Biringer (Boston College)|
''Growth of Betti numbers and a probabilistic take on Gromov Hausdorff convergence'']]
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|-
|-
|October 19
| Peng Gao (Simons Center for Geometry and Physics)
| [[#Peng Gao (Simons Center for Geometry and Physics)|
''string theory partition functions and geodesic spectrum'']]
|[http://www.math.wisc.edu/~bwang/ Wang]
|-
|-
|October 26
|Oct. 25
| [http://www.math.wisc.edu/~nelson/ Jo Nelson] (Wisconsin)
|Emily Stark (Utah)
| [[#Jo Nelson (Wisconsin) |
| Action rigidity for free products of hyperbolic manifold groups
''Cylindrical contact homology as a well-defined homology theory? Part I'']]
|(Dymarz)
| local
|-
|-
|November 2
|Nov. 8
| [http://www.bowdoin.edu/~jtaback/ Jennifer Taback] (Bowdoin)
|Max Forester (University of Oklahoma)
| [[#Jennifer Taback (Bowdoin)|
|Spectral gaps for stable commutator length in some cubulated groups
''The geometry of twisted conjugacy classes in Diestel-Leader groups'']]
|(Dymarz)
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|-
|-
|November 9
|Nov. 22
| [http://math.uchicago.edu/~wilsonj/ Jenny Wilson] (Chicago)
|Yu Li (Stony Brook University)
| [[#Jenny Wilson (Chicago)|
|On the structure of Ricci shrinkers
''FI-modules for Weyl groups'']]
|(Huang)
| [http://www.math.wisc.edu/~ellenber/ Ellenberg]
|-
|-
|November 16
|[http://www.math.uic.edu/people/profile?id=GasJ574 Jonah Gaster] (UIC)
|[[#Jonah Gaster (UIC)|
''A Non-Injective Skinning Map with a Critical Point'']]
|[http://www.math.wisc.edu/~rkent/ Kent]
|-
| Thanksgiving Recess
|
|
|
|-
|November 30
| [http://www.its.caltech.edu/~shinpei/ Shinpei Baba] (Caltech)
|[[#Shinpei Baba (Caltech)|
''Grafting and complex projective structures'']]
|[http://www.math.wisc.edu/~rkent/ Kent]
|-
|December 7
| [http://math.uchicago.edu/~mann/ Kathryn Mann] (Chicago)
|[[#Kathryn Mann (Chicago)|
''The group structure of diffeomorphism groups'']]
|[http://www.math.wisc.edu/~rkent/ Kent]
|-
|
|}
|}


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


===Owen Sizemore (Wisconsin)===
===Xiangdong Xie===
''Operator Algebra Techniques in Measureable Group Theory''


Measurable group theory is the study of groups via their actions on measure spaces. While the classification for amenable groups was essentially complete by the early 1980's, progress for nonamenable groups has been slow to emerge. The last 15 years has seen a surge in the classification of ergodic actions of nonamenable groups, with methods coming from diverse areas. We will survey these new results, as well as, give an introduction to the operator algebra techniques that have been used.
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.


===Mireille Boutin (Purdue)===
===Kuang-Ru Wu===
''The Pascal Triangle of a discrete Image: definition, properties, and application to object segmentation''


We define the Pascal Triangle of a discrete (gray scale) image as a pyramidal ar-
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.
rangement of complex-valued moments and we explore its geometric significance. In
particular, we show that the entries of row k of this triangle correspond to the Fourier
series coefficients of the moment of order k of the Radon transform of the image. Group
actions on the plane can be naturally prolonged onto the entries of the Pascal Triangle. We study the induced action of some common group actions, such as translation,
rotations, and reflections, and we propose simple tests for equivalence and self-
equivalence for these group actions. The motivating application of this work is the
problem of recognizing ”shapes” on images, for example characters, digits or simple
graphics. Application to the MERGE project, in which we developed a fast method for segmenting hazardous material signs on a cellular phone, will be also discussed.  


This is joint work with my graduate students Shanshan Huang and Andrew Haddad.
===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.  


===Ben Schmidt (Michigan State)===
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.
''Three manifolds of constant vector curvature.''


A Riemannian manifold M is said to have extremal curvature K if all sectional curvatures are bounded above by K or if all sectional curvatures are bounded below by K.  A manifold with extremal curvature K has constant vector curvature K if every tangent vector to M belongs to a tangent plane of curvature K.  For surfaces, having constant vector curvature is equivalent to having constant curvature.  In dimension three, the eight Thurston geometries all have constant vector curvature.  In this talk, I will discuss the classification of closed three manifolds with constant vector curvature.  Based on joint work with Jon Wolfson.
===Karin Melnick===


===Ian Biringer (Boston College)===
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.
''Growth of Betti numbers and a probabilistic take on Gromov Hausdorff convergence''


We will describe an asymptotic relationship between the volume and the Betti numbers of certain locally symmetric spaces. The proof uses an exciting new tool: a synthesis of Gromov-Hausdorff convergence of Riemannian manifolds and Benjamini-Schramm convergence from graph theory.
===Joerg Schuermann===


===Peng Gao (Simons Center for Geometry and Physics)===
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.
''string theory partition functions and geodesic spectrum''


String theory partition functions often have nice modular properties, which is well understood within the context of representation theory of (supersymmetric extensions) of Virasoro algebra.
===David Massey===
However, many questions of physical importance are preferrably addressed when string theory is formulated in terms of non-linear sigma model on a Riemann surface with a Riemannian manifold as target space. Traditionally, physicists have studied such sigma models within the realm of perturbation theory, overlooking a large class of very natural critical points of the path integral, namely, closed geodesics on the target space Riemannian manifold. We propose how to take into account the effect of these critical points on the path integral, and initiate its study on Ricci flat targe spaces, such as the K3 surface.


===Jo Nelson (Wisconsin)===
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.
''Cylindrical contact homology as a well-defined homology theory? Part I''


In this talk I will define all the concepts in the title, starting with what a contact manifold is.  I will also  explain how the heuristic arguments sketched in the literature since 1999 fail to define a homology theory and provide a foundation for a well-defined cylindrical contact homology, while still providing an invariant of the contact structure.  A later talk will provide us with a large class of examples under which one can compute a well-defined version of cylindrical contact homology via a new approach the speaker developed for her thesis that is distinct and completely independent of previous specialized attempts.
===Antoine Song===


===Jennifer Taback (Bowdoin)===
TBA
''The geometry of twisted conjugacy classes in Diestel-Leader groups''


The problem of computing the Reidemsieter number R(f)  of a group automorphism f, that is, the number of f-twisted conjugacy classes, is related to questions in Lefschetz-Nielsen fixed point theory.  We say a group has property R-infinity if every group automorphism has infinitely many twisted conjugacy classes.  This property has been studied by Fel’shtyn, Gonzalves, Wong, Lustig, Levitt and others, and has applications outside of topology.
==Fall Abstracts==
Twisted conjugacy classes in lamplighter groups are well understood both geometrically and algebraically.  In particular the lamplighter group L_n does not have property R-infinity iff (n,6)=1. In this talk I will extend these results to Diestel-Leader groups with a surprisingly different conclusion.  The family of Diestel-Leader groups provides a natural geometric generalization of the lamplighter groups.  I will define these groups, as well as Diestel-Leader graphs and describe how these results include a computation of the automorphism group of this family.
This is joint work with Melanie Stein and Peter Wong.


===Jenny Wilson (Chicago)===
===Ruobing Zhang===
''FI-modules for Weyl groups''


Earlier this year, Church, Ellenberg, and Farb developed a new framework for studying sequences of representations of the symmetric groups, using a concept they call an FI--module. I will give an overview of this theory, and describe how it generalizes to sequences of representations of the classical Weyl groups in Type B/C and D. The theory of FI--modules has provided a wealth of new results by numerous authors working in algebra, geometry, and topology. I will outline some of these results, including applications to configurations spaces and groups related to the braid group.
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.


===Jonah Gaster (UIC)===
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.
''A Non-Injective Skinning Map with a Critical Point''


Following Thurston, certain classes of 3-manifolds yield holomorphic maps on the Teichmuller spaces of their boundary components. Inspired by numerical evidence of Kent and Dumas, we present a negative result about the regularity of such maps. Namely, we construct a path of deformations of the hyperbolic structure on a genus-2 handlebody, with two rank-1 cusps. The presence of some extra symmetry yields information about the convex core, which is used to conclude some inequalities involving the extremal length of a certain symmetric curve family. The existence of a critical point for the associated skinning map follows.
===Emily Stark===


===Shinpei Baba (Caltech)===
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.
''Grafting and complex projective structures''


A complex projective structure is a certain geometric structure on a (real) surface, and it corresponds a representation from the fundamental group of the base surface into PSL(2,C).  We discuss about a certain surgery operation, called a 2&pi;&ndash;grafting, which produces a different projective structure, preserving its holonomy representation.
===Max Forester===
This surgery is closely related to three-dimensional hyperbolic geometry.
 
===Kathryn Mann (Chicago)===
''The group structure of diffeomorphism groups''
 
Abstract:
What is the relationship between manifolds and the structure of their
diffeomorphism groups?
On the positive side, a remarkable theorem of Filipkiewicz says that the
group structure determines the manifold: if Diff(M) and Diff(N) are
isomorphic, then M and N are diffeomorphic.
On the negative side, we know little else.  Could the group Diff(M) act by
diffeomorphisms on M in nonstandard ways?  Does the "size" of Diff(M) say
anything about the complexity of M?  Ghys asked if M and N are manifolds,
and the group of compactly supported diffeomorphisms of N injects into the
group of compactly supported diffeomorphisms of M, can the dimension of M
be less than dim(N)?  We'll discuss these and other questions, and answer
these in the (already quite rich) case of dim(M)=1.


I will discuss stable commutator length (scl) in groups, and some gap theorems for the scl spectrum. Such results say that for various groups, scl of an element is always either zero or is larger than some uniform constant. I will discuss the cases of right-angled Artin groups and certain right-angled Coxeter groups. This is joint work with Pallavi Dani, Ignat Soroko, and Jing Tao.


===Yu Li===
We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As an application, we show that any Ricci shrinker whose second eigenvalue of the curvature operator is positive must be a quotient of sphere.


== Archive of past Geometry seminars ==
== Archive of past Geometry seminars ==
 
2018-2019  [[Geometry_and_Topology_Seminar_2018-2019]]
<br><br>
2017-2018 [[Geometry_and_Topology_Seminar_2017-2018]]
<br><br>
2016-2017  [[Geometry_and_Topology_Seminar_2016-2017]]
<br><br>
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]
<br><br>
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]
<br><br>
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]
<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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