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Geometry & Topology Seminar 2009-10

 

Term 1 2009/10

Thursday October 8, 15:00, room B3.01

Iain Aitchison (Melbourne)

The hype and spin of natural sphere eversion

Abstract: Smale proved in 1957 that an embedded 2-sphere in R3 can be smoothly everted, that is, turned inside out by continuous deformation through a family of smooth immersions (regularly homotopy). The first public explicit pictures of how this might be done emerged in 1966, based on Boy's immersion of the projective plane in R3. Another visual proof was animated in the late 1970's, based on Morin's symmmetric immersion of the sphere; a remake, using the Willmore flow, was created in 1998 by Sullivan et al -- `The Optiverse' -- winning a prize at the 1998 ICM. On the other hand, Thurston developed another approach in the early 1970's, based on twisted ribbons, which was animated in the movie `Outside In'. These movies can be viewed on YouTube.

Beautiful and intriguing as they are, none of these approaches yields an understanding of the eversion which is both completely conceptual mathematically, and moreover, such that the actual eversion can be visually grasped, at every stage.

We present a new such eversion at two levels: the first basic level, using a (2,-1) torus knot rotating on an unknotted solid donut, and the simplest possible immersed disc with a single arc/clasp intersection, completely describes the eversion from embedded sphere to embedded sphere (inside out). (This can be completely described in one written page. A (simple) POVRAY animation of this process has been partially completed.)

The second level describes the mathematical origins of the eversion: this combines the most basic features of each of: the diffeomorphisms SO(3) &cong RP3 &cong T1S2 &cong L(2,-1), and the natural Seifert fibering coming from the 2-fold cover Spin(3) &cong S3 &cong SU(2), via the Hopf fibration originating with C2 &cong R4. Spin structures and hyperspace give the title of the talk, and underly it: the eversion explicitly arises from the `-1'-Dehn surgery description of S3.


Thursday October 15, 15:00, room B3.01

Vladimir Markovic (Warwick)

Nearly geodesic immersed surfaces in hyperbolic three manifolds and the dynamics of the frame flow

Abstract: I will discuss my recent work with Jeremy Kahn showing that one can immerse many nearly geodesic closed surfaces in a given closed hyperbolic 3-manifold. This result is interesting to topologists because such surfaces are essential in the given 3-manifold. However in our construction we only use basic hyperbolic geometry and the mixing of the frame flow that acts on the 2-frame bundle over the hyperbolic 3-manifold.


Thursday October 22, 15:00, room B3.01

Clifford Earle (Cornell)

Earle-Marden coordinates in genus two: an example

Abstract: This example is a special case of joint work with Al Marden. In the example we consider a compact Riemann surface of genus two that has been pinched along a simple closed geodesic to produce a pair of tori joined at a singular point. That singularity can be opened up by a plumbing construction.

We model that plumbing by a Kleinian group construction, and the deformation theory of Kleinian groups yields holomorphic coordinates for an augmented Teichmueller space.

All this will be explained at the talk. We shall also show that the resulting coordinates (in this example) are independent, up to isomorphism, of the Kleinian group chosen for the construction.


Thursday October 29, 15:00, room B3.01

Martin Bridson (Oxford)

Actions of mapping class groups on spaces of non-positive curvature

Abstract: I'll sketch the proof of several results that constrain the way in which mapping class groups and automorphism groups of free groups can act by isometries on CAT(0) spaces. I shall discuss consequences concerning the linear representations of these groups as well as maps between them.


Thursday November 5, 15:00, room B3.01

Alex Suciu (Northeastern)

The Alexander polynomial and its friends

Abstract: The classical Alexander polynomial from knot theory admits many generalizations, all based on the idea of extracting information about a space from the homology of its abelian covers. In this talk, I will discuss some connections between the (multi-variable) Alexander polynomial, the cohomology jumping loci, and the BNS invariants of a finitely generated group.


Thursday November 12, 15:00, room B3.01

Joan Porti (Barcelona)

Twisted cohomology of hyperbolic 3-manifolds

Abstract: I will discuss a vanishing theorem of Raghunathan for compact hyperbolic 3-manifolds, and how it generalizes to the finite volume case.

The holonomy representation of an oriented hyperbolic three manifold lifts to SL(2,C), and its composition with the nth-symmetric power of the standard action of SL(2,C) on C2 defines an action on Cn+1. We are interested in the cohomology of the manifold twisted by this representation, for n larger or equal to one. When n=2, C3 is identified with the lie algebra sl(2,C) and this is the infinitesimal rigidity theorem of Weil and Matsushima-Murakami (compact case) and Garland (finite volume case). These cohomology computations may be used to define Reidemeister torsions.

This is joint work with Pere Menal-Ferrer.


Thursday November 19, 15:00, room B3.01

Stephan Wehrli (Paris)

Colored Khovanov homology and sutured Floer homology

Abstract: The relationship between categorifications of quantum knot polynomials and Floer homology invariants is intriguing and still poorly-understood. In this talk, I will discuss a connection between Khovanov's categorification of the reduced n-colored Jones polynomial and sutured Floer homology, a relative version of Heegaard Floer homology recently developed by Andras Juhasz. As an application, I will prove that Khovanov's categorification detects the unknot when n > 1.

This is joint work with J. Elisenda Grigsby.


Thursday November 26, 15:00, room B3.01

Ken Baker (Miami)

Rational open books, cabling, and contact structures

Abstract: The Giroux Correspondence is a one-to-one correspondence between contact structures up to isotopy and open book decompositions up to positive stabilization. An open book decomposition of a 3-manifold is a link with a fibration of its exterior such that each fiber is a Seifert surface for the link. Cabling a link component produces a new open book decomposition (with few exceptions). We will describe how the contact structure supported by an open book changes under cabling, generalizing Hedden's result for open books in S^3. We'll also define rational open books and discuss their cablings.

This is joint work with John Etnyre and Jeremy Van Horn-Morris.


Thursday December 3, 15:00, room B3.01

Raphael Zentner (Muenster)

A vanishing result for a Casson type instanton invariant over negative definite four-manifolds

Abstract: After a historical introduction to the Casson invariant and after reviewing instanton gauge theory (Donaldson theory) we shall come to speak about Casson type invariants on 4-manifolds. In particular, we shall focus on those defined on negative definite 4-manifolds as suggested by Teleman. We shall present results on these among which a vanishing result and its possible applications.


Thursday December 10, 15:00, room B3.01

Shinpei Baba (Bonn)

Grafting operations on complex projective structures

Abstract: A (complex) projective structure is a certain geometric structure on a (closed) surface. This structure comes with a holonomy representation of the surface group into PSL(2,C), which does not need to be discrete or faithful. In addition, such a holonomy representation corresponds to infinitely many different projective structures on the surface.

(2π-)grafting is a certain surgery operation on a projective structure that produces different projective structures, preserving its holonomy representation. Gallo-Kapovich-Marden (2000) asked whether, given two projective structures with the same holonomy representation, there is a sequence of graftings and inverse-graftings that transforms one to the other.

We answer this question affirmatively for all purely loxodromic representations, which are generic in the representation variety.

Term 2 2009/10

Thursday January 14, 15:00, room B1.01

Stephen Tawn (Warwick)

The Hilden subgroup of the braid group

Abstract: The Hilden, or Wicket, subgroup of the braid group on 2n string can be defined in several ways. As the group of motions of n hoops in halfspace, as the stabiliser of n hoops under the action of the braid group on tangles, or as a mapping class group. Using the mapping class point of view, we will construct a simply-connected complex with an action by the Hilden group. We will then use this action to show that the Hilden group has a finite presentation.


Thursday January 21, 15:00, room B1.01

Anne Thomas (Oxford)

Lattices with surface subgroups

Abstract: Let I_{p,q} be Bourdon's building, the unique simply-connected 2-complex such that each 2-cell is a regular right-angled hyperbolic p-gon, and the link at each vertex is the complete bipartite graph K_{q,q}. We determine all triples (p,q,g) such that the automorphism group of I_{p,q} admits a lattice Γ so that the quotient of I_{p,q} by Γ is a compact surface of genus g. In these cases, the surface group naturally embeds in Γ. This is joint work with David Futer.


Thursday January 28, 15:00, room B1.01

Alexander Coward (Oxford)

Upper bounds on Reidemeister moves

Abstract: (Joint work with Marc Lackenby.) Given any two diagrams of the same knot or link, we provide an explicit upper bound on the number of Reidemeister moves required to pass between them in terms of the number of crossings in each diagram. This provides a new and conceptually simple solution to the equivalence problem for knot and links.


Thursday February 4, 15:00, room B1.01

Andrei Tetenov (Gorno-Altaisk)

The structure and rigidity of self-similar Jordan arcs

Abstract: Suppose the Jordan arc γ is the invariant set for a digraph system S of contraction similarities. Then either the arc γ is an invariant set for some multizipper and admits non-trivial deformations or γ is a straight line segment, the system S does not satisfy the weak separation property, and the self-similar structure (γ,S) is rigid.


Thursday February 11, 15:00, room B1.01

Dmitri Panov (Imperial)

Hyperbolic geometry and symplectic manifolds with c1=0

Abstract: We will show how to use 4-dimensional hyperbolic geometry to construct symplectic manifolds of dimension 6 with c1=0. In particular using Davis manifold we construct a first known example of simply connected symplectic 6-fold with c1=0 that does not admit a compatible Kahler structure. If the time permits we will describe further examples with arbitrary large Betti numbers together with higher-dimensional analogues of this construction, producing non-Kahler symplectic Fano manifolds. This is a joint work with Joel Fine.


Thursday February 18, 15:00, room B1.01

Hee Jung Kim (Max-Planck)

Double point surgery and configurations of surfaces in 4-manifolds

Abstract: We introduce a new operation, double point surgery on a configuration of surfaces in a 4-manifold, and use it to construct configurations that are smoothly knotted, without changing the topological type or the smooth embedding type of the individual components of the configuration. Taking branched covers, we produce smoothly exotic actions of Zm ⊕ Zn on simply connected 4-manifolds with complicated fixed-point sets.


Thursday February 25, 15:00, room B1.01

Andrzej Zuk (Paris)

L2 Betti numbers of closed manifolds

Abstract: TBA


Thursday March 4, 15:00, room B1.01

Caroline Series (Warwick)

Top terms of trace polynomials in Kra's plumbing construction

Abstract: (Joint work with Sara Maloni.) Kra's plumbing construction manufactures a surface S by `plumbing' together a suitable family of triply punctured spheres. This gives a natural pants decomposition of S, together with a projective structure for which the associated holonomy representation ρ depends on the `plumbing parameters' τi. In particular Trace ρ(γ), for γ ∈ π1(S), is a polynomial in the τi.

Simple curves on S can be described in terms of their Dehn-Thurston coordinates relative to the pants decomposition. After explaining the construction, we show that if γ is simple there is a very simple formula relating the coefficients of the top terms of ρ(γ) and its Dehn-Thurston coordinates.

Maloni spoke in last year's seminar: in this talk we present a combinatorial proof which applies to an arbitrary pants decomposition and which involves a rather interesting result on matrix products.


Thursday March 11, 15:00, room B1.01

Daniele Alessandrini (Strasbourg)

On the compactification of Teichmuller-like parameter spaces

Abstract: The construction of compactifications of Teichmuller spaces in the approach of Morgan and Shalen has close relationships with tropical geometry. By studying the general properties of the logarithmic limit set of real semi-algebraic sets, it is possible to generalize their construction and to understand some of its properties. When applied to Teichmuller spaces, this gives the compactification of Thurston, and the natural piecewise linear structure of the boundary appears automatically, showing clearly how this structure is related with the semi-algebraic structure of the interior part. It is also possible to construct a compactification of the parameter spaces of strictly convex projective structures on a closed n-manifold. In this case objects from tropical geometry also appear in the interpretation of the boundary points.


Thursday March 11, 16:15, room B3.03

Jim Howie (Heriot-Watt)

Dehn surgery and prime factors

Abstract: A 3-manifold obtained by Dehn surgery on a knot is generically - but not always - irreducible. The Cabling Conjecture of Gonzalez-Acuna and Short asserts that a reducible manifold can be obtained only in a specified way, in which case the number of prime factors is precisely 2. I shall discuss the weaker conjecture that Dehn surgery can never produce a manifold with more than 2 prime factors.


Thursday March 18, 15:00, room D1.07

Laurent Bartholdi (Göttingen)

Amenability of groups and algebras

Abstract: Amenability of groups (a notion introduced by von Neumann in his study of the Banach-Tarski paradox) is a far-reaching generalization of "finiteness". It leads to a fundamental dichotomy of the "landscape" of groups, and I will describe some features of amenable / non-amenable groups, providing typical or fundamental examples along the way.

I will explain interesting related questions in the world of algebras, and give an application to the theory of cellular automata.

 

Term 3 2009/10

Thursday May 13, 14:00, room B3.03

Andrei Tetenov (Gorno-Altaisk)

The bending locus for Kleinian groups in space

Abstract: Let Λ be the limit set of a discontinuous group G of Möbius authomorphisms of 4-ball B4, and let C(Λ) be the hyperbolic convex hull of Λ. We consider elementary geometrical and topological properties of the bending locus BL of ∂C(Λ). We prove that if G is a group generated by a finite number of reflections then the quotient space BL/G ⊂ ∂C(Λ)/G is a finite union of totally geodesic surfaces Si, with boundary ∂ Si consisting of a finite number of closed geodesics. We also claim that the 1-skeleton BL(1) of the bending locus for a convex cocompact group G is nowhere dense in ∂C(Λ).


Thursday May 13, 15:30, room B3.03

Elmas Irmak (Bowling Green)

Mapping class groups and complexes of curves on orientable/nonorientable surfaces

Abstract: I will talk about the relation between the mapping class groups of surfaces and the automorphism groups and the superinjective simplicial maps of the complexes of curves on surfaces for both orientable and nonorientable surfaces. I will also talk about the proof that each superinjective simplicial map of the complex of curves of a compact, connected, nonorientable surface is induced by a homeomorphism of the surface if g + n is at least 5 or at most 3, where g is the genus and n is the number of boundary components of the surface.


Thursday May 20, 15:00, room MS.04

Stephane Lamy (Warwick)

Normal subgroups in the Cremona group

Abstract: The Cremona group is the group of birational transformations of the plane. Following a construction by Manin it is possible to see this group as acting by isometries on an infinite dimensional hyperbolic space. Using these ideas we are able to produce many example of proper normal subgroups in the Cremona group. (Joint work with Serge Cantat)


Thursday May 20, 16:30, room MS.04

Oyku Yurttas (Liverpool)

Dynnikov coordinates and pseudo-Anosov braids

Abstract: Isotopy classes of orientation preserving homeomorphims on a finitely punctured disk are represented by braids. In this talk I will present a method for computing the dilatation of pseudo-Anosov braids using the Dynnikov coordinate system which is computationally much more efficient than the usual Thurston train track aproach. If time permits, I will talk about the relation between Dynnikov matrices and the train track transition matrix with an illustrative example that also shows the local dynamics around the unstable foliation in the boundary of Teichmuller space.


Thursday May 27, 15:00, room MS.04

Ivan Smith (Cambridge)

Floer cohomology and pencils of quadrics

Abstract: There is a classical relationship, in algebraic geometry, between a hyperelliptic curve and an associated pencil of quadric hypersurfaces. We investigate symplectic aspects of this relationship and their consequences in low-dimensional topology.


Thursday June 3, 15:00, room MS.04

Maciej Borodzik (Warsaw)

Morse theory for singular complex curves in C^2 and signatures of torus knots

Abstract: For a given complex curve C in ℂ2 and a generic point z, we study the links S(z,r) ∩ C. Here S(z,r) denotes the standard 3-sphere in ℂ2 with centre at z and radius r. As r varies, the links can change. We describe what happens when r crosses a singular point of C. Studying the changes of Tristram-Levine signatures of S(z,r) ∩ C, we obtain deep information about the topology and singularities of C.


Thursday June 10, 15:00, room MS.04

Nicolas Bergeron (Curie)

Torsion in the homology of 3-manifolds

Abstract: Compact 3-manifolds can have "a lot" of torsion in their homology. I will try to quantify what "a lot" means and sketch the proofs of two different kind of results: growth of torsion in abelian covers, and growth of torsion for some twisted local systems in residually finite covers. This is joint work with Akshay Venkatesh.


Thursday July 1, 15:30, room MS.04

David Futer (Temple)

The geometry of unknotting tunnels

Abstract: Given a knot K in S3, an unknotting tunnel for K is an arc τ from K to K, such that the complement of K and τ is a genus two handlebody. Fifteen years ago, Colin Adams asked a series of questions about how unknotting tunnels fit into the hyperbolic structure on the knot complement. For example: is τ isotopic to a geodesic? Can it be arbitrarily long, relative to a maximal cusp neighborhood? Does τ appear as an edge in the Epstein-Penner polyhedral decomposition?

Although the most general versions of these questions are still open today, I will describe fairly complete answers in the special case where K is created by a ``generic'' Dehn filling. As an application, there is an explicit family of knots in S3 whose tunnels are arbitrarily long. This is joint work with Daryl Cooper and Jessica Purcell.