Geometry and Topology Seminar 2019-2020

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The Geometry and Topology seminar meets in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm.
For more information, contact Richard Kent.

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Fall 2013

date speaker title host(s)
September 6
September 13, 10:00 AM in 901! Alex Zupan (Texas) Totally geodesic subgraphs of the pants graph Kent
September 20
September 27
October 4
October 11
October 18 Jayadev Athreya (Illinois) Gap Distributions and Homogeneous Dynamics Kent
October 25 Joel Robbin (Wisconsin) GIT and [math]\displaystyle{ \mu }[/math]-GIT local
November 1 Anton Lukyanenko (Illinois) Uniformly quasi-regular mappings on sub-Riemannian manifolds Dymarz
November 8 Neil Hoffman (Melbourne) Verified computations for hyperbolic 3-manifolds Kent
November 15 Khalid Bou-Rabee (Minnesota) On generalizing a theorem of A. Borel Kent
November 22 Morris Hirsch (Wisconsin) Common zeros for Lie algebras of vector fields on real and complex

2-manifolds.

local
Thanksgiving Recess
December 6 Sean Paul (Wisconsin) (Semi)stable Pairs I local
December 13 Sean Paul (Wisconsin) (Semi)stable Pairs II local

Fall Abstracts

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]\displaystyle{ \mu }[/math]-GIT

Many problems in differential geometry can be reduced to solving a PDE of form

[math]\displaystyle{ \mu(x)=0 }[/math]

where [math]\displaystyle{ x }[/math] ranges over some function space and [math]\displaystyle{ \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]\displaystyle{ G }[/math] acts on a projective algebraic variety [math]\displaystyle{ X }[/math], then the complexification [math]\displaystyle{ G^c }[/math] also acts and there is an isomorphism of orbifolds

[math]\displaystyle{ X^s/G^c=X//G:=\mu^{-1}(0)/G }[/math]

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]\displaystyle{ \mathbb{R}^3 }[/math]. To help generalize this paradox, Borel proved the following result on free groups.

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

What is the class, [math]\displaystyle{ \mathcal{L} }[/math], of groups that may play the role of [math]\displaystyle{ F }[/math] in Borel’s Theorem? Since the free group of rank two is in [math]\displaystyle{ \mathcal{L} }[/math], it follows that all residually free groups are in [math]\displaystyle{ \mathcal{L} }[/math]. In this talk, we present some methods for determining whether a finitely generated group is in [math]\displaystyle{ \mathcal{L} }[/math]. Using these methods, we give a concrete example of a finitely generated group in [math]\displaystyle{ \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]\displaystyle{ X }[/math] on a manifold [math]\displaystyle{ M }[/math] has nonempty zero set [math]\displaystyle{ Z(X) }[/math], provided [math]\displaystyle{ M }[/math] is compact with empty boundary and [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ Z(X) }[/math] is compact, [math]\displaystyle{ i(X) }[/math] denotes the intersection number of [math]\displaystyle{ X }[/math] with the zero section of the tangent bundle.

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

• Assume [math]\displaystyle{ X }[/math] is analytic, [math]\displaystyle{ Z(X) }[/math] is compact and [math]\displaystyle{ i(X) \neq 0 }[/math]. Let [math]\displaystyle{ \mathfrak{g} }[/math] be generated by analytic vector fields [math]\displaystyle{ Y }[/math] on [math]\displaystyle{ M }[/math] such that the vectors [math]\displaystyle{ [X,Y]p }[/math] and [math]\displaystyle{ Xp }[/math] are linearly dependent at all [math]\displaystyle{ p \in M }[/math]. Then [math]\displaystyle{ 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

date speaker title host(s)
January 24
January 31 Spencer Dowdall (UIUC) Fibrations and polynomial invariants for free-by-cyclic groups Kent
February 7
February 14
February 21 Ioana Suvaina (Vanderbilt) ALE Ricci flat Kahler surfaces from a Tian-Yau construction approach Maxim
February 28 Jae Choon Cha (POSTECH, Korea) Universal bounds for the Cheeger-Gromov rho-invariants Maxim
March 7 Mustafa Kalafat (Michigan-State and Tunceli) Conformally Kahler Surfaces and Orthogonal Holomorphic Bisectional Curvature
March 14
Spring Break
March 28
April 4 Matthew Kahle (Ohio) MOVED TO COLLOQUIUM SLOT Dymarz
April 11 Yongqiang Liu (UW-Madison and USTC-China) Nearby cycles and Alexander modules of hypersurface complements Maxim
April 18 Pallavi Dani (LSU) Large-scale geometry of right-angled Coxeter groups. Dymarz
April 25 Jingzhou Sun (Stony Brook) On the Demailly-Semple jet bundles of hypersurfaces in the 3-dimensional complex projective space Wang
May 2
May 9

Spring Abstracts

Spencer Dowdall (UIUC)

Fibrations and polynomial invariants for free-by-cyclic groups

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

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

Ioana Suvaina (Vanderbilt)

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 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)

Universal bounds for the Cheeger-Gromov rho-invariants"

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.

Mustafa Kalafat (Michigan-State and Tunceli)

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.

Matthew Kahle (Ohio)

TBA

Yongqiang Liu

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.

Pallavi Dani (LSU)

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)

"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. 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

date speaker title host(s)
Monday, August 18, 2:25 in 901! David Epstein (Warwick University) Machine Learning and Topology Robbin


Summer Abstracts

David Epstein (Warwick University)

Machine Learning and Topology

Modern scientists, particularly biologists, have to deal with datasets that live in high-dimensional spaces. A typical image has 1000 x 1000 pixels, and each pixel has an real-valued intensity, so that we can regard the image as a point in the space R^1,000,000. 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 dimension 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- dimensional progression, getting steadily worse. It might be possible to detect this deterioration with a sequence of images made from blood samples. This progression can be modelled as a curve, so 1-dimensional, in R^1,000,000. 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 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 R^n. 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 R^n 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 project the point cloud onto the submanifold, and examine its properties in a space of dimension k rather in dimension n. This approach to dimension 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

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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