Geometry and Topology Seminar 2011-2012

Fall 2011

The seminar will be held in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm

date	speaker	title	host(s)
September 9	Gloria Mari Beffa (UW Madison)	The pentagram map and generalizations: discretizations of AGD flows	[local]
September 16	Ke Zhu (University of Minnesota)	Thin instantons in G2-manifolds and Seiberg-Witten invariants	Yong-Geun
September 23	Antonio Ache (UW Madison)	Obstruction-Flat Asymptotically Locally Euclidean Metrics	[local]
September 30	John Mackay (Oxford University)	What does a random group look like?	Tullia
October 7	David Fisher (Indiana University)	Hodge-de Rham theory for infinite dimensional bundles and local rigidity	Richard and Tullia
October 14	Erwan Lanneau (University of Marseille, CPT)	Dilatations of pseudo-Anosov homeomorphisms and Rauzy-Veech induction	Jean Luc
October 21	Ruifang Song (UW Madison)	The Picard-Fuchs equations of Calabi-Yau hypersurfaces in partial flag varieties	[local]
October 24 ( with Geom. analysis seminar)	Valentin Ovsienko (University of Lyon)	The pentagram map and generalized friezes of Coxeter	Gloria
November 4	Steven Simon (NYU)	Equivariant Analogues of the Ham Sandwich Theorem	Max
November 18	Igor Zelenko (Texas A&M University)	On geometry of curves of flags of constant type	Gloria
December 1 at 2 PM in Ingraham 114	Bing Wang (Simons Center for Geometry and Physics)	Uniformization of algebraic varieties. NOTE SPECIAL PLACE AND TIME: Thursday, December 1 at 2 PM in Ingraham 114.	[Jeff]
December 2	David Dumas (University of Illinois at Chicago)	Real and complex boundaries in the character variety	Richard
December 9	Brian Clarke (Stanford)	TBA	Jeff

Abstracts

Gloria Mari Beffa (UW Madison)

The pentagram map and generalizations: discretizations of AGD flows

GIven an n-gon one can join every other vertex with a segment and find the intersection of two consecutive segments. We can form a new n-gon with these intersections, and the map taking the original n-gon to the newly found one is called the pentagram map. The map's properties when defined on pentagons are simple to describe (it takes its name from this fact), but the map turns out to have a unusual number of other properties and applications.

In this talk I will give a quick review of recent results by Ovsienko, Schwartz and Tabachnikov on the integrability of the pentagram map and I will describe on-going efforts to generalize the pentagram map to higher dimensions using possible connections to Adler-Gelfand-Dikii flows. The talk will NOT be for experts and will have plenty of drawings, so come and join us.

Ke Zhu (University of Minnesota)

Thin instantons in G2-manifolds and Seiberg-Witten invariants

For two nearby disjoint coassociative submanifolds $C$ and $C'$ in a $G_2$-manifold, we construct thin instantons with boundaries lying on $C$ and $C'$ from regular $J$-holomorphic curves in $C$. It is a high dimensional analogue of holomorphic stripes with boundaries on two nearby Lagrangian submanifolds $L$ and $L'$. We explain its relationship with the Seiberg-Witten invariants for $C$. This is a joint work with Conan Leung and Xiaowei Wang.

Antonio Ache (UW Madison)

Obstruction-Flat Asymptotically Locally Euclidean Metrics

Given an even dimensional Riemannian manifold [math]\displaystyle{ (M^{n},g) }[/math] with [math]\displaystyle{ n\ge 4 }[/math], it was shown in the work of Charles Fefferman and Robin Graham on conformal invariants the existence of a non-trivial 2-tensor which involves [math]\displaystyle{ n }[/math] derivatives of the metric, arises as the first variation of a conformally invariant and vanishes for metrics that are conformally Einstein. This tensor is called the Ambient Obstruction tensor and is a higher dimensional generalization of the Bach tensor in dimension 4. We show that any asymptotically locally Euclidean (ALE) metric which is obstruction flat and scalar-flat must be ALE of a certain optimal order using a technique developed by Cheeger and Tian for Ricci-flat metrics. We also show a singularity removal theorem for obstruction-flat metrics with isolated [math]\displaystyle{ C^{0} }[/math]-orbifold singularities. In addition, we show that our methods apply to more general systems. This is joint work with Jeff Viaclovsky.

John Mackay (Oxford University)

What does a random group look like?

Twenty years ago, Gromov introduced his density model for random groups, and showed when the density parameter is less than one half a random group is, with overwhelming probability, (Gromov) hyperbolic. Just as the classical hyperbolic plane has a circle as its boundary at infinity, hyperbolic groups have a boundary at infinity which carries a canonical conformal structure.

In this talk, I will survey some of what is known about random groups, and how the geometry of a hyperbolic group corresponds to the structure of its boundary at infinity. I will outline recent work showing how Pansu's conformal dimension, a variation on Hausdorff dimension, can be used to give a more refined geometric picture of random groups at small densities.

David Fisher (Indiana University)

Hodge-de Rham theory for infinite dimensional bundles and local rigidity

It is well known that every cohomology class on a manifold can be represented by a harmonic form. While this fact continues to hold for cohomology with coefficients in finite dimensional vector bundles, it is also fairly well known that it fails for infinite dimensional bundles. In this talk, I will formulate a notion of a harmonic cochain in group cohomology and explain what piece of the cohomology can be represented by harmonic cochains. I will use these ideas to prove a vanishing theorem that motivates a family of generalizations of property (T) of Kazhdan. If time permits, I will discuss applications to local rigidity of group actions.

Erwan Lanneau (University of Marseille, CPT)

Dilatations of pseudo-Anosov homeomorphisms and Rauzy-Veech induction

In this talk I will explain the link between pseudo-Anosov homeomorphisms and Rauzy-Veech induction. We will see how to derive properties on the dilatations of these homeomorphisms (I will recall the definitions) and as an application, we will use the Rauzy-Veech-Yoccoz induction to give lower bound on dilatations. This is a common work with Corentin Boissy (Marseille).

Ruifang Song (UW Madison)

The Picard-Fuchs equations of Calabi-Yau hypersurfaces in partial flag varieties

We introduce a system of differential equations associated to a smooth algebraic variety X acted by a complex Lie group G and a G-linearlized line bundle L on X. We show that this system is holonomic and thus its solution space is finite dimensional assuming G acts on X with finitely many orbits. When X is a partial flag variety, we show that this system gives the Picard-Fuchs system of Calabi-Yau hypersurfaces in X. When X is a toric variety, our construction recovers GKZ systems and extended GKZ systems, which play important roles in studying periods of Calabi-Yau hypersurfaces in toric varieties. This is based on joint work with Bong Lian and Shing-Tung Yau.

Valentin Ovsienko (University of Lyon)

The pentagram map and generalized friezes of Coxeter

The pentagram map is a discrete integrable system on the moduli space of n-gons in the projective plane (which is a close relative of the moduli space of genus 0 curves with n marked points). The most interesting properties of the pentagram map is its relations to the theory of cluster algebras and to the classical integrable systems (such as the Boussinesq equation). I will talk of the recent results proving the integrability as well as of the algebraic and arithmetic properties of the pentagram map. In particular, I will introduce the space of 2-frieze patterns generalizing that of the classical Coxeter friezes and define the structure of cluster manifold on this space. The talk is based on joint works with Sophie Morier-Genoud, Richard Schwartz and Serge Tabachnikov.

Steven Simon (NYU)

Equivariant Analogues of the Ham Sandwich Theorem

The Ham Sandwich Theorem, one of the earliest applications of algebraic topology to geometric combinatorics, states that under generic conditions any n finite Borel measures on R^n can be bisected by a single hyperplane. Viewing this theorem as a Z_2-symmetry statement for measures, we generalize the theorem to other finite groups, notably the finite subgroups of the spheres S^1 and S^3, in each case realizing group symmetry on Euclidian space as group symmetries of its Borel measures by studying the topology of associated spherical space forms. Direct equipartition statements for measures are given as special cases. We shall also discuss the contributions of the tangent bundles of these manifolds in answering similar questions.

Igor Zelenko (Texas A&M University)

On geometry of curves of flags of constant type

The talk is devoted to the (extrinsic) geometry of curves of flags of a vector space $W$ with respect to the action of a subgroup $G$ of the $GL(W)$. We develop an algebraic version of Cartan method of equivalence or an analog of Tanaka prolongation for such problem. Under some natural assumptions on the subgroup $G$ and on the flags, one can pass from the filtered objects to the corresponding graded objects and describe the construction of canonical bundles of moving frames for these curves in the language of pure Linear Algebra. The scope of applicability of the theory includes geometry of natural classes of curves of flags with respect to reductive linear groups or their parabolic subgroups. As simplest examples, this includes the projective and affine geometry of curves.

Our motivation to study such equivalence problems comes from the new approach to the geometry of structures of nonholonomic nature on manifolds such as vector distributions, sub-Riemannian structure etc. This approach is based on the Optimal Control Theory and it consists of the reduction of the equivalence problem for such nonholonomic geometric structures to the (extrinsic) differential geometry of curves in Lagrangian Grassmannians and, more generally, of curves of flags of isotropic and coisotropic subspaces in a linear symplectic space with respect to the action of the Linear Symplectic Group. The application of the general theory to the geometry of such curves case will be discussed in more detail.

Bing Wang (Simons Center for Geometry and Physics)

Uniformization of algebraic varieties

For algebraic varieties of general type with mild singularities, we show the Bogmolov-Yau inequality holds. If equality is attained, then this variety is a global quotient of complex hyperbolic space away from a subvariety. This will be a more detailed version of the speaker's colloquium talk.

David Dumas (University of Illinois at Chicago)

Real and complex boundaries in the character variety

The set of holonomy representations of complex projective structures on a compact Riemann surface is a submanifold of the SL(2,C) character variety of the fundamental group. We will discuss the real- and complex-analytic geometry of this manifold and its interaction with the Morgan-Shalen compactification of the character variety. In particular we show that the subset consisting of holonomy representations that extend over a given hyperbolic 3-manifold group (of which the surface is an incompressible boundary) is discrete.

Brian Clarke (Stanford)

Ricci Flow, Analytic Stability, and the Space of Kähler Metrics

I will consider the space of all Kähler metrics on a fixed, compact, complex manifold as a submanifold of the manifold of all Riemannian metrics. The geometry induced on it in this way coincides with a Riemannian metric first defined by Calabi in the 1950s. After giving a detailed study of the Riemannian distance function - in particular determining the completion of the space of Kähler metrics - I will give a new analytic stability criterion for the existence of a Kähler--Einstein metric on a Fano manifold in terms of the Ricci flow and the distance function. Additionally, I will describe a result showing that the Kähler--Ricci flow converges as soon as it converges in the very weak metric sense. This is joint work with Yanir Rubinstein.

Fall-2010-Geometry-Topology

Spring 2012

The seminar will be held in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm

date	speaker	title	host(s)
January 27	Tullia Dymarz (UW Madison)	Geometry of solvable Lie groups and quasi-isometric rigidity	[local]
February 3	Laurentiu Maxim (UW Madison)	L2-Betti numbers of affine hypersurface complements	[local]
February 17	Talia Fernos (University of North Carolina and the Hebrew University of Jerusalem)	Property (T), its friends, and its adversaries	Tullia
April 13	Greg Friedman (TCU)	Intersection Homology and Stratified Spaces	Max
April 20	Karin Melnick (University of Maryland)	Normal Forms for Conformal Lorentzian Vector Fields	Tullia
April 27	Will Cavendish (Princeton)	TBA	Richard

Abstracts

Tullia Dymarz (UW Madison)

Geometry of solvable Lie groups and quasi-isometric rigidity

Gromov's program on classifying finitely generated groups up to quasi-isometry was initiated with his polynomial growth theorem. With this theorem he showeed that the class of (virtually) nilpotent groups is closed under quasi-isometries. One of the current active projects in this area, started by Eskin-Fisher-Whyte, is the classification of lattices in solvable Lie groups up to quasi-isometry. I will give an overview of the geometry of solvable Lie groups along with a sketch of how to use this geometry to prove quasi-isometric rigidity.

Laurentiu Maxim (UW Madison)

L²-Betti numbers of hypersurface complements

I will present vanishing results for the L²-cohomology of complements to complex affine hypersurfaces in general position at infinity.

Talia Fernos (University of North Carolina and the Hebrew University of Jerusalem)

Property (T), its friends, and its adversaries

Property (T) has many successes in the study of a broad range of areas: von Neumann Algebras, dynamics, and geometric group theory to name a few. It is a property of groups that was introduced by Kazhdan in 1967; he showed that all higher rank lattices share this property. All property (T) groups are finitely generated. This is a key ingredient in Margulis' Arithmeticity Theorem for higher rank lattices. Simply stated, property (T) is a type of deformation rigidity for unitary representations. Namely, unitary representation which are close to containing the trivial representation actually do.

Property (T) can be seen as an analytical (versus geometric) property: it is an invariant under measure equivalence but not under quasi isometry. Property (T) groups do not admit non-trivial actions on many "simple" spaces. Such spaces include trees, the circle (with a sufficiently smooth action), and walled spaces. In this talk we will give a survey of property (T) and related properties such as the Haagerup property.

Greg Friedman (TCU)

Intersection Homology and Stratified Spaces

Intersection homology was developed by Goresky and MacPherson in order to extend Poincare duality and related invariants from manifold theory to "singular spaces", such as non-smooth algebraic varieties. We'll provide an introduction to these topics, concluding with some recent work and work in progress concerning bordism groups of singular spaces and symmetric signatures of singular spaces.

Karin Melnick (University of Maryland)

Normal Forms for Conformal Lorentzian Vector Fields

Isometries of a Riemannian or pseudo-Riemannian manifold fixing a point are conjugate to their differential via the exponential map. No such linearization exists in general for conformal transformations fixing a point. The main theorem of this talk asserts that on a real-analytic Lorentzian manifold M, any conformal vector field vanishing at a point has linearizable flow, or M is conformally flat. This result leads to a normal form for any such vector field near its singularity. (Joint work with Charles Frances.)

Will Cavendish (Princeton)

"Towers of Covering Spaces of 3-manifolds and Mapping Solenoids"

Agol's recent resolution of the virtualy Haken conjecture together with work of Wise and the tameness theorem show that any $\pi_1$-injective map $f:S\to M$ of a surface $S$ into a geometric 3-manifold $M$ can be lifted to a map $\tilde{f}$ into a finite sheeted covering space $\widetilde{M}\to M$ that is homotopic to an embedding. Examples of Rubinstein and Wang show, however, that there exist maps of surfaces into non-hyperbolic 3-manifolds that are not ``virtually embedded" in this sense. In this talk we will discuss a construction called the mapping solenoid of $f$, and show how the cohomology groups of this object can be viewed as obstructions to solving topological lifting problems. We will then discuss the cohomology of the mapping solenoid associated to the Rubinstein-Wang examples and show that these obstructions do not vanish in this setting.

Fall-2010-Geometry-Topology