Geometry and Topology Seminar

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.