Geometry and Topology Seminar

Thursday January 13, 15:00, room MS.04

Michael Weiss (Aberdeen)

Vassiliev’s discriminant method: some simplifications and extensions

Abstract: Vassiliev’s discriminant method is a method for calculating the cohomology of certain spaces of smooth maps f:M -> R^k, where M is typically a closed manifold and the smooth maps f are subject to local conditions (only so-and-so singularities) or multi-local conditions. It makes very original use of Alexander duality, viewing the space of maps in question as an open subset of an infinite dimensional vector space, intersecting that with finite dimensional affine subspaces and going for the locally finite homology of the complement (hence “discriminant method”). It also relies very much on interpolation. The discriminant method is closely related to “Vassiliev invariants” in knot theory, but it should not be viewed as a low-dimensional tool only.

“Simplifications” means: more use of real semi-algebraic geometry, less use of random methods like transversality, hence more obvious permission to Alexander duality theorems. “Extensions”: this refers to the case of multi-local conditions where functor calculus language can be used to formulate a reasonably clear statement. The simplifications and extensions are joint work with Rui Reis. I like the discriminant method because it helps me to develop an analogue of Morse theory with 2-dimensional target (k=2).

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

James Anderson (Southampton)

Separating geodesics in hyperbolic 3-manifolds

Abstract: In the 1970s, Horowitz and Randol showed that on any surface S of negative Euler characteristic, there are n-tuples of closed geodesics c₁, ..., c_n whose lengths are equal for any hyperbolic structure on S. Characterizing such n-tuples has proven to be very difficult.

There is a partial converse, due originally to McShane, which states that if c is a simple closed geodesic on S, then the length of c, as a function on the Teichmüller space of S, determines the conjugacy class of c. We will discuss an extension of this partial converse to compact hyperbolizable 3-manifolds.

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

TBA (TBA)

TBA

Abstract: TBA

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

Mary Rees (Liverpool)

The Mating Construction in Complex Dynamics

Abstract: The mating construction was invented by Adrien Douady and John Hubbard in the 1980's. It is a way of describing the dynamics of some rational maps in terms of the dynamics of two polynomials. It is quite a general construction. For example, it is valid for a positive proportion of critically finite rational maps in degree two (and quite possibly in any degree). The construction not only describes the dynamics of individual rational maps, but also appears to give pictures of parts of parameter space. In some cases these pictures are valid. Positive results are known when one of the polynomials in the mating is star-like, the so-called rabbit polynomial, for example. In other cases, shared mating, the representation of a rational map by a mating in more than one way, means that not all the pictures of parameter space which are suggested by the mating pictures can be interpreted as simply as might at first be thought. Study of matings with the aeroplane polynomial gives ample indication of the complications that arise.

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

Nicholas Touikan (Oxford)

Finding group splittings

Abstract:I will present an algorithm which given as input --- (a) a finite presentation < X | R > for a group without 2-torsion, (b) a solution to the word problem with respect to this presentation and (c) a positive integer k --- outputs a finite collection t₁,...,t_n of "tracks" with the property that if the group < X | R > admits a k-acylindrical geometric splitting, then up to automorphism, an edge group of this splitting is carried by one of these tracks.

I will define what all this means and give some applications, namely the detection of non-trivial free decompositions and the detection of parabolic splittings of relatively hyperbolic groups.

Thursday February 17, 15:00, room MS.04

Yankı Lekili (Cambridge)

Quilted Floer homology of 3-manifolds

Abstract: We introduce quilted Floer homology (QFH), a new invariant of 3-manifolds equipped with an indefinite circle valued Morse function (i.e. broken fibration). This is yet another localization of Seiberg-Witten theory and a natural extension of Perutz's 4-manifold invariants associated with broken Lefschetz fibrations, making it a (3+1) theory. We relate Perutz's theory to Heegaard Floer theory by giving an isomorphism between QFH and HF+ for extremal spin^c structures with respect to the fibre of the Morse function. As applications, we give new computations of Heegaard Floer homology and a characterization of sutured Floer homology.

Thursday February 24, 15:00, room MS.04

Caroline Series (Warwick)

Continuous motions of limit sets

Abstract: A Kleinian group is a discrete group of isometries of hyperbolic 3-space. Its limit set is the set of points where orbits accumulate on the boundary. We show that if a sequence of geometrically finite groups converges strongly to a geometrically finite limit, then the limit sets converge uniformly. We also discuss what happens when the convergence is not strong or the groups are not geometrically finite. Joint work with Mahan Mj.

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

Juan Souto (Michigan)

Homomorphisms between mapping class groups

Abstract: Suppose that X and Y are surfaces of finite topological type, X with at least genus six and Y with genus less than twice the genus of X. In this talk I will describe all homomorphism between the mapping class group of X to the mapping class group of Y. This is join work with Javier Aramayona.

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

Adam Epstein (Warwick)

Transversality principles in holomorphic dynamics

Abstract: The moduli space of all degree D rational maps is an orbifold of dimension 2D - 2. We present a language for describing dynamically natural subspaces, for example, the loci of maps having either (*) specified critical orbit relations, (*) cycles of specified period and multiplier, (*) parabolic cycles of specified degeneracy and index, (*) Herman ring cycles of specified rotation number, or some combination thereof.

We present a methodology for proving the smoothness and transversality of such loci. The natural setting for the discussion is a family of deformation spaces arising functorially from first principles in Teichmüller theory. Transversality flows from an infinitesimal rigidity principle (following Thurston), in the corresponding variational theory viewed cohomologically (following Kodaira-Spencer). Results for deformation spaces may then be transferred to moduli space. Moreover, the deformation space formalism and associated transversality principles apply more generally to finite type transcendental maps.

Thursday March 17, 15:00, room MS.04

Owen Cotton-Barratt (Oxford)

Detecting geometry in finite quotients

Abstract: The profinite completion of a discrete group is an object which captures the structure of its finite quotients. We will introduce this and consider the following problem: how similar must two groups be if they have the same profinite completion? We will recall the definition of the ends of a group, and the associated structure theory. We will then use this to explain new work giving conditions in which the number of ends of a group can be detected in its profinite completion.