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. I shall discuss consequences concerning the linear representations of these groups as well as maps between them.Ībstract: 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.
#TRIPLY TWISTIT MOBIUS SPACE FREE#
We shall also show that the resulting coordinates (in this example) are independent, up to isomorphism, of the Kleinian group chosen for the construction.Īctions of mapping class groups on spaces of non-positive curvatureĪbstract: 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.
We model that plumbing by a Kleinian group construction, and the deformation theory of Kleinian groups yields holomorphic coordinates for an augmented Teichmueller space.Īll this will be explained at the talk. That singularity can be opened up by a plumbing construction. 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. 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.Įarle-Marden coordinates in genus two: an exampleĪbstract: This example is a special case of joint work with Al Marden. This result is interesting to topologists because such surfaces are essential in the given 3-manifold.
Nearly geodesic immersed surfaces in hyperbolic three manifolds and the dynamics of the frame flowĪbstract: 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. Spin structures and hyperspace give the title of the talk, and underly it: the eversion explicitly arises from the `-1'-Dehn surgery description of S 3. The second level describes the mathematical origins of the eversion: this combines the most basic features of each of: the diffeomorphisms SO(3) &cong RP 3 &cong T 1S 2 &cong L(2,-1), and the natural Seifert fibering coming from the 2-fold cover Spin(3) &cong S 3 &cong SU(2), via the Hopf fibration originating with C 2 &cong R 4.
A (simple) POVRAY animation of this process has been partially completed.) (This can be completely described in one written page. 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). These movies can be viewed on YouTube.īeautiful 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.
#TRIPLY TWISTIT MOBIUS SPACE MOVIE#
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'. 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. The first public explicit pictures of how this might be done emerged in 1966, based on Boy's immersion of the projective plane in R 3.
The hype and spin of natural sphere eversionĪbstract: Smale proved in 1957 that an embedded 2-sphere in R 3 can be smoothly everted, that is, turned inside out by continuous deformation through a family of smooth immersions (regularly homotopy).
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