Geometry and Topology Seminar 2019-2020: Difference between revisions

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


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




== Fall 2013==
== Spring 2020 ==
 
 


{| cellpadding="8"
{| cellpadding="8"
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!align="left" | host(s)
!align="left" | host(s)
|-
|-
|September 6
|Feb. 7
|
|Xiangdong Xie  (Bowling Green University)
|
| Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|
|(Dymarz)
|-
|September 13, <b>10:00 AM in 901!</b>
| [http://www.ma.utexas.edu/users/zupan/ Alex Zupan] (Texas)
| [[#Alex Zupan (Texas)| ''Totally geodesic subgraphs of the pants graph'']]
| [http://www.math.wisc.edu/~rkent/ Kent]
|-
|September 20
|
|
|
|-
|September 27
|
|
|
|-
|October 4
|
|
|
|-
|October 11
|
|
|
|-
|October 18
| [http://www.math.uiuc.edu/~jathreya/ Jayadev Athreya] (Illinois)
|[[#Jayadev Athreya (Illinois)| ''Gap Distributions and Homogeneous Dynamics'']]
| [http://www.math.wisc.edu/~rkent/ Kent]
|-
|October 25
| [http://www.math.wisc.edu/~robbin/ Joel Robbin (Wisconsin)]
| [[#Joel Robbin (Wisconsin) | ''GIT and <math>\mu</math>-GIT'']]
| local
|-
|-
|November 1
|Feb. 14
| [http://lukyanenko.net/ Anton Lukyanenko (Illinois)]
|Xiangdong Xie  (Bowling Green University)
| [[#Anton Lukyanenko (Illinois)| ''Uniformly quasi-regular mappings on sub-Riemannian manifolds'']]
| Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|(Dymarz)
 
|-
|-
|November 8
|Feb. 21
| Neil Hoffman (Melbourne)
|Xiangdong Xie  (Bowling Green University)
| [[#Neil Hoffman (Melbourne)| ''Verified computations for hyperbolic 3-manifolds'']]
| Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
|[http://www.math.wisc.edu/~rkent/ Kent]
|(Dymarz)
|-
|-
|November 15
|Feb. 28
| Khalid Bou-Rabee (Minnesota)
|Kuang-Ru Wu (Purdue University)
| [[#Khalid Bou-Rabee (Minnesota)| ''On generalizing a theorem of A. Borel'']]
|Griffiths extremality, interpolation of norms, and Kahler quantization
|[http://www.math.wisc.edu/~rkent/ Kent]
|(Huang)
|-
|-
|November 22
|Mar. 6
| Morris Hirsch (Wisconsin)
|Yuanqi Wang (University of Kansas)
| [[#Morris Hirsch (Wisconsin)| ''Common zeros for Lie algebras of vector fields on real and complex
|Moduli space of G2−instantons on 7−dimensional product manifolds
2-manifolds.'']]
|(Huang)
| local
|-
|-
|Thanksgiving Recess
|Mar. 13 <b>CANCELED</b>
|  
|Karin Melnick (University of Maryland)
|
|A D'Ambra Theorem in conformal Lorentzian geometry
|
|(Dymarz)
|-
|-
|December 6
|<b>Mar. 25</b> <b>CANCELED</b>
| Sean Paul (Wisconsin)
|Joerg Schuermann (University of Muenster, Germany)
| [[#Sean Paul (Wisconsin)| ''(Semi)stable Pairs I'']]
|An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles
| local
|(Maxim)
|-
|-
|December 13
|Mar. 27 <b>CANCELED</b>
| Sean Paul (Wisconsin)
|David Massey (Northeastern University)
| [[#Sean Paul (Wisconsin)| ''(Semi)stable Pairs II'']]
|Extracting easily calculable algebraic data from the vanishing cycle complex
| local
|(Maxim)
|-
|-
|
|<b>Apr. 10</b> <b>CANCELED</b>
|Antoine Song (Berkeley)
|TBA
|(Chen)
|}
|}


== Fall Abstracts ==
== Fall 2019 ==
 
===Alex Zupan (Texas)===
''Totally geodesic subgraphs of the pants graph''
 
Abstract:
For a compact surface S, the associated pants graph P(S) consists of vertices corresponding to pants decompositions of S and edges corresponding to elementary moves between pants decompositions.  Motivated by the Weil-Petersson geometry of Teichmüller space, Aramayona, Parlier, and Shackleton conjecture that the full subgraph G of P(S) determined by fixing a multicurve is totally geodesic in P(S).  We resolve this conjecture in the case that G is a product of Farey graphs.  This is joint work with Sam Taylor.
 
===Jayadev Athreya (Illinois)===
''Gap Distributions and Homogeneous Dynamics''
 
Abstract:
We discuss the notion of gap distributions of various lists of numbers in [0, 1], in particular focusing on those which are associated to certain low-dimensional dynamical systems. We show how to explicitly compute some examples using techniques of homogeneous dynamics, generalizing earlier work on gaps between Farey Fractions. This works gives some possible notions of `randomness' of special trajectories of billiards in polygons, and is based partly on joint works with J. Chaika, J. Chaika and S. Lelievre, and with Y.Cheung. This talk may also be of interest to number theorists.
 
===Joel Robbin (Wisconsin)===
GIT and  <math>\mu</math>-GIT
 
Many problems in differential geometry can be reduced to solving a PDE of form
<br><br>
<math>
    \mu(x)=0
</math>
<br><br>
where <math>x</math> ranges over some function space and <math>\mu</math> is an infinite dimensional analog of the moment map in symplectic geometry. 
In Hamiltonian dynamics the moment map was introduced to use a group action to reduce the number of degrees of freedom in the ODE.
It was soon discovered that the moment map could be applied to Geometric Invariant Theory:
if a  compact Lie group <math>G</math> acts on a projective algebraic variety <math>X</math>,
then the complexification <math>G^c</math> also acts and there is an isomorphism of orbifolds
<br><br>
<math>
    X^s/G^c=X//G:=\mu^{-1}(0)/G
</math>
<br><br>
between the space of orbits of Mumford's stable points and the Marsden-Weinstein quotient.
 
In September of 2013 Dietmar Salamon, his student Valentina Georgoulas, and I wrote an exposition of (finite dimensional) GIT from the point of view of symplectic geometry.
The theory works for compact Kaehler manifolds, not just projective varieties.
I will describe our paper in this talk; the following Monday Dietmar will give more details in the Geometric Analysis Seminar.
 
===Anton Lukyanenko (Illinois)===
''Uniformly quasi-regular mappings on sub-Riemannian manifolds''
 
Abstract:
A quasi-regular (QR) mapping between metric manifolds is a branched cover with bounded dilatation, e.g. f(z)=z^2. In a joint work with K. Fassler and K. Peltonen, we define QR mappings of sub-Riemannian manifolds and show that:
1) Every lens space admits a uniformly QR (UQR) mapping f.
2) Every UQR mapping leaves invariant a measurable conformal structure.
The first result uses an explicit "conformal trap" construction, while the second builds on similar results by Sullivan-Tukia and a connection to higher-rank symmetric spaces.
 
===Neil Hoffman (Melbourne)===
''Verified computations for hyperbolic 3-manifolds''
 
Abstract:
Given a triangulated 3-manifold M a natural question is: Does M admit a hyperbolic structure?
 
While this question can be answered in the negative if M is known to
be reducible or toroidal, it is often difficult to establish a
certificate of hyperbolicity, and so computer methods have developed
for this purpose. In this talk, I will describe a new method to
establish such a certificate via verified computation and compare the
method to existing techniques.
 
This is joint work with Kazuhiro Ichihara, Masahide Kashiwagi,
Hidetoshi Masai, Shin'ichi Oishi, and Akitoshi Takayasu.
 
===Khalid Bou-Rabee (Minnesota)===
''On generalizing a theorem of A. Borel''
 
The proof of the Hausdorff-Banach-Tarski paradox relies on the existence of a nonabelian free group in the group of rotations of <math>\mathbb{R}^3</math>. To help generalize this paradox, Borel proved the following result on free groups.
 
Borel’s Theorem (1983): Let <math>F</math> be a free group of rank two. Let <math>G</math> be an arbitrary connected semisimple linear algebraic group (i.e., <math>G = \mathrm{SL}_n</math> where <math>n \geq 2</math>). If <math>\gamma</math> is any  nontrivial element in <math>F</math> and <math>V</math> is any proper subvariety of <math>G(\mathbb{C})</math>, then there exists a homomorphism <math>\phi: F \to G(\mathbb{C})</math> such that <math>\phi(\gamma) \notin V</math>.
 
What is the class, <math>\mathcal{L}</math>, of groups that may play the role of <math>F</math> in Borel’s Theorem? Since the free group of rank two is in <math>\mathcal{L}</math>, it follows that all residually free groups are in <math>\mathcal{L}</math>. In this talk, we present some methods for determining whether a finitely generated group is in <math>\mathcal{L}</math>. Using these methods, we give a concrete example of a finitely generated group in <math>\mathcal{L}</math> that is *not* residually free. After working out a few other examples, we end with a discussion on how this new theory provides an answer to a question of Brueillard, Green, Guralnick, and Tao concerning double word maps. This talk covers joint work with Michael Larsen.
 
===Morris Hirsch (Wisconsin)===
''Common zeros for Lie algebras of  vector fields on real and complex 2-manifolds.''
 
The celebrated Poincare-Hopf theorem states that a vector field <math>X</math> on a manifold
<math>M</math> has nonempty zero set <math>Z(X)</math>, provided <math>M</math> is compact with empty boundary and
<math>M</math> has nonzero Euler characteristic. Surprising little is known about the set of
common zeros of two or more vector fields, especially when <math>M</math> is not compact.
One of the few results in this direction is a remarkable theorem of Christian
Bonatti (Bol. Soc. Brasil. Mat. 22 (1992), 215–247), stated below. When <math>Z(X)</math> is
compact, <math>i(X)</math> denotes the intersection number of <math>X</math> with the zero section of the
tangent bundle.
 
<math>\cdot </math> Assume <math> dim_{\mathbb{R}(M)} ≤ 4</math>, <math>X</math> is analytic, <math>Z(X)</math> is compact and <math>i(X) \neq 0</math>. Then
every analytic vector field commuting with <math>X</math> has a zero in <math>Z(X)</math>.
In this talk I will discuss the following analog of Bonatti’s theorem. Let <math>\mathfrak{g}</math> be
a Lie algebra of analytic vector fields on a real or complex 2-manifold <math>M</math>, and set
<math>Z(g) := \cap_{Y \in \mathfrak{g}} Z(Y)</math>.
 
• Assume <math>X</math> is analytic, <math>Z(X)</math> is compact and <math>i(X) \neq 0</math>. Let <math>\mathfrak{g}</math> be generated by
analytic vector fields <math>Y</math> on <math>M</math> such that the vectors <math>[X,Y]p</math> and <math>Xp</math> are linearly
dependent at all <math>p \in M</math>. Then <math>Z(\mathfrak{g}) \cap Z(X) \neq \emptyset </math>.
Related results on Lie group actions, and nonanalytic vector fields, will also be
treated.
 
===Sean Paul (Wisconsin)===
''(Semi)stable Pairs I''
 
===Sean Paul (Wisconsin)===
''(Semi)stable Pairs II''
 
 
== Spring 2014 ==
 


{| cellpadding="8"
{| cellpadding="8"
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!align="left" | host(s)
!align="left" | host(s)
|-
|-
|January 24
|Oct. 4
|  
|Ruobing Zhang (Stony Brook University)
|
| Geometric analysis of collapsing Calabi-Yau spaces
|
|(Chen)
|-
|-
|January 31
|[http://www.math.uiuc.edu/~dowdall/ Spencer Dowdall (UIUC)]
|[[#Spencer Dowdall (UIUC)| ''Fibrations and polynomial invariants for free-by-cyclic groups'']]
|[http://www.math.wisc.edu/~rkent Kent]
|
|-
|-
|February 7
|Oct. 25
|  
|Emily Stark (Utah)
|
| Action rigidity for free products of hyperbolic manifold groups
|
|(Dymarz)
|-
|-
|February 14
|Nov. 8
|  
|Max Forester (University of Oklahoma)
|
|Spectral gaps for stable commutator length in some cubulated groups
|
|(Dymarz)
|-
|-
|February 21
|Nov. 22
| [http://www.math.vanderbilt.edu/~suvaini/ Ioana Suvaina (Vanderbilt)]
|Yu Li (Stony Brook University)
| [[#Ioana Suvaina (Vanderbilt)| ''ALE Ricci flat Kahler surfaces from a Tian-Yau construction approach'']]
|On the structure of Ricci shrinkers
| [http://www.math.wisc.edu/~maxim/ Maxim]
|(Huang)
|
|-
|February 28
|[http://gt.postech.ac.kr/~jccha/ Jae Choon Cha (POSTECH, Korea)]
|[[#Jae Choon Cha (POSTECH)| ''Universal bounds for the Cheeger-Gromov rho-invariants'']]
|[http://www.math.wisc.edu/~maxim Maxim]
|
|-
|March 7
| Mustafa Kalafat (Michigan-State and Tunceli)
|[[#Mustafa Kalafat (Michigan-State and Tunceli)| ''Conformally Kahler Surfaces and Orthogonal Holomorphic Bisectional Curvature'']]
|
|-
|March 14
|
|
|
|-
|Spring Break
|
|
|
|-
|March 28
|
|
|
|-
| April 4
| [http://matthewkahle.org/ Matthew Kahle (Ohio)]
| [[#Matthew Kahle (Ohio)| ''MOVED TO COLLOQUIUM SLOT'']]
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|-
|April 11
| Yongqiang Liu (UW-Madison and USTC-China)
|[[#Yongqiang Liu| ''Nearby cycles and Alexander modules of hypersurface complements'']]
|[http://www.math.wisc.edu/~maxim/ Maxim]
|-
| April 18
| [https://www.math.lsu.edu/~pdani/ Pallavi Dani (LSU)]
| [[#Pallavi Dani (LSU)| ''Large-scale geometry of right-angled Coxeter groups.'']]
|[http://www.math.wisc.edu/~dymarz/ Dymarz]
|-
|April 25
| [http://www.math.sunysb.edu/~jsun/ Jingzhou Sun (Stony Brook)]
| [[#Jingzhou Sun(Stony Brook)| ''On the Demailly-Semple jet bundles of hypersurfaces in the 3-dimensional complex projective space'']]
|[http://www.math.wisc.edu/~bwang Wang]
|-
|May 2
|
|
|
|-
|May 9
|
|
|
|-
|-
|}
|}


== Spring Abstracts ==
==Spring Abstracts==
 
===Spencer Dowdall (UIUC)===
''Fibrations and polynomial invariants for free-by-cyclic groups''


The beautiful theory developed by Thurston, Fried and McMullen provides a near complete picture of the various ways a hyperbolic 3-manifold M can fiber over the circle. Namely, there are distinguished convex cones in the first cohomology M^1(M;R) whose integral points all correspond to fibrations of M, and the dynamical features of these fibrations are all encoded by McMullen's "Teichmuller polynomial."
===Xiangdong Xie===


This talk will describe recent work developing aspects of this picture in the setting of a free-by-cyclic group G. Specifically, I will introduce a polynomial invariant that determines a convex polygonal cone C in the first cohomology of G whose integral points all correspond to algebraically and dynamically interesting splittings of G. The polynomial invariant additionally provides a wealth of dynamical information about these splittings. This is joint work with Ilya Kapovich and Christopher J. Leininger.
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.


===Ioana Suvaina (Vanderbilt)===
===Kuang-Ru Wu===
''ALE Ricci flat Kahler surfaces from a Tian-Yau construction approach"


The talk presents an explicit  classification of the ALE Ricci flat Kahler surfaces (M,J,g), generalizing
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.
previous classification results of Kronheimer. The manifolds are related to Q-Gorenstein deformations
of quotient singularities of type C^2/G, with G a finite subgroup of U(2).  
Using this classification, we show how these metrics can also be obtained by a construction of Tian-Yau.
In particular, we find good compactifications of the underlying complex manifold M.


===Jae Choon Cha (POSTECH)===
===Yuanqi Wang===
''Universal bounds for the Cheeger-Gromov rho-invariants"
$G_{2}-$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.


Cheeger and Gromov showed that there is a universal bound of their L2 rho-invariants of a fixed smooth closed (4k-1)-manifold, using a deep analytic method.  We give a new topological proof of the existence of a universal bound.  For 3-manifolds, we give explicit estimates in terms of triangulations, Heegaard splittings, and surgery descriptions.  The proof employs interesting ideas including controlled chain homotopy and a geometric reinterpretation of the Atiyah-Hirzebruch bordism spectral sequence.  Applications include new results on the complexity of 3-manifolds.
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.


===Mustafa Kalafat (Michigan-State and Tunceli)===
===Karin Melnick===
''Conformally Kahler Surfaces and Orthogonal Holomorphic Bisectional Curvature''


We show that a compact complex surface which admits a conformally Kahler metric g of positive orthogonal holomorphic bisectional curvature is biholomorphic to the complex projective plane. In addition, if g is a Hermitian metric which is Einstein, then the biholomorphism can be chosen to be an isometry via which g becomes a multiple of the Fubini-Study metric. This is joint work with C.Koca.
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.


===Matthew Kahle (Ohio)===
===Joerg Schuermann===
''TBA''


===Yongqiang Liu===
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.
''Nearby cycles and Alexander modules of hypersurface complements''


For a polynomial transversal at infinity, we show that the Alexander modules of the hypersurface complement can be realized by the nearby cycle complex, and we obtain a divisibility result for the associated Alexander polynomial.  As an application, we use nearby cycles to recover the mixed Hodge structure on the torsion Alexander modules, as defined by  Dimca and Libgober.
===David Massey===


===Pallavi Dani (LSU)===
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.
''A finitely generated group can be endowed with a natural metric which
is unique up to coarse isometries, or quasi-isometries. A fundamental
question is to classify finitely generated groups up to
quasi-isometry.  I will report on the progress on this question in the
case of right-angled Coxeter groups.  In particular I will describe
how topological features of the visual boundary can be used to
classify a family of hyperbolic right-angled Coxeter groups. I will
also discuss the connection with commensurability, an algebraic
property which implies quasi-isometry, but is stronger in general.
This is joint work with Anne Thomas.''


===Jingzhou Sun (Stony Brook)===
===Antoine Song===
"On the Demailly-Semple jet bundles of hypersurfaces in the 3-dimensional complex projective space"


Let X be a smooth hypersurface of degree d in the 3-dimensional complex projective space.
TBA
By totally algebraic calculations, we prove that on the third Demailly-Semple jet bundle X_3 of X,
the Demailly-Semple line bundle is big for d not ness than 11,
and that on the fourth  Demailly-Semple jet bundle X_4 of X,
the Demailly-Semple line bundle is big for d not ness than 10,  improving a recent result of Diverio.


== Summer 2014 ==
==Fall Abstracts==


{| cellpadding="8"
===Ruobing Zhang===
!align="left" | date
!align="left" | speaker
!align="left" | title
!align="left" | host(s)
|-
|<b>Monday, August 18, 2:25 in 901!</b>
| David Epstein (Warwick University)
|[[#David Epstein (Warwick University)| ''Machine Learning and Topology'']]
| [http://www.math.wisc.edu/~robbin/ Robbin]
|-
|}


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.


== Summer Abstracts ==
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.


===David Epstein (Warwick University)===
===Emily Stark===
''Machine Learning and Topology''


Modern scientists, particularly biologists, have to deal with datasets that
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.
live in high-dimensional spaces. A typical image has 1000 X1000 pixels, and
each pixel has an real-valued intensity, so that we can regard the image as a
point in the space R1;000;000. The objective of a lot of modern research is to
find ways to drastically reduce the dimension from a million to a dimension
that human brains are capable of understanding|ideally this means to di-
mension 1 or 2, or, reluctantly, dimension 3, but any reduction in dimension
is helpful.


Suppose, for example, there is a disease that typically shows a one-
===Max Forester===
dimensional progression, getting steadily worse. It might be possible to de-
tect this deterioration with a sequence of images made from blood samples.
This progression can be modelled as a curve, so 1-dimensional, in R1;000;000.
Stochastic factors are always present in biological measurements. So the
model would consist of a probability distribution that clusters in the vicinity
of a curve.


How might one find (an approximation to) the curve, given only the point
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.
cloud in the higher dimensional euclidean space? More generally, suppose
that the point cloud is clustered round a patch of surface (dimension 2) or a k-
dimensional non-linear patch in Rn. How can one recover (an approximation
to) the patch? More generally still (more mathematically complete, but
further from biological applications), given a point cloud in Rn that clusters
round a compact k-dimensional submanifold, possibly with boundary, how
might one find (an approximation to) the submanifold?


If one succeeds in finding the k-dimensional submanifold, one can then
===Yu Li===
project the point cloud onto the submanifold, and examine its properties
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.
in a space of dimension k rather in dimension n. This approach to dimen-
sion reduction will be applicable to only some point clouds, and completely
different techniques will be applicable in different cases.
 
The talk will describe some partial progress towards achieving the above
objectives, with a sketch plan for further progress. Manifold learning is a
topic being worked on by hundreds of researchers, and, as an outsider, I am
not claiming originality. I would be interested to learn of others following
similar lines of investigation.
 
A main tool is the use of (multi-dimensional) splines.


== 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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