Geometry and Topology Seminar

Thursday Jan 15, 15:15, room B3.02

John Jones (Warwick)

String Topology

Abstract: String homology was introduced by Moira Chas and Dennis Sullivan in 1999. Their idea was to do intersection theory on the loop space of a finite dimensional manifold. In a subsequent paper, published in 2003, Ralph Cohen and myself gave a different approach to the theory using the general methods of algebraic topology and homotopy theory. In this talk I will give an introduction to string homology, and emphasize two fundamental problems.

1. How does one calculate string homology?

2. What exactly does string homology depend on? One of the key features of string homology is that it does depend on Poincare duality in the underlying manifold. Is it possible that string homology gives a source of invariants which distinguish between closed manifolds that are homotopy equivalent but not homeomorphic?

Thursday Jan 22, 15:15, room B3.02

Brendan Owens (Glasgow)

Torus knot surgeries and negative-definite four-manifolds

Abstract: (Joint work with Saso Strle.) Given a knot K in the three-sphere, one may ask which Dehn surgeries on K bound negative-definite four-manifolds. I will explain how this question is of interest in both knot theory and contact topology. I will answer the question for the case of torus knots. This involves two constructions of four-manifolds and an obstruction using Donaldson's diagonalisation theorem.

Thursday Feb 5, 15:15, room B3.02

Ciprian Manolescu (Cambridge/UCLA)

Symplectic instanton homology

Abstract: Floer's instanton homology was originally defined as an invariant of integral homology 3-spheres. The Atiyah-Floer Conjecture claims that there should be a symplectic counterpart to instanton theory, based on Lagrangian Floer homology. Starting from a Heegaard decomposition of a 3-manifold, I will explain one way to make sense of the symplectic side of the Atiyah-Floer conjecture, for arbitrary 3-manifolds. This is joint work with Chris Woodward.

Thursday Feb 5, 16:30, room B3.02

Taehee Kim (Konkuk)

Concordance of knots and Alexander polynomials

Abstract: In this talk, I will start with the background and brief history of concordance. The relationship between concordance and Alexander polynomials will be discussed. In particular, I will talk about concordance of two knots when those two knots have coprime Alexander polynomials.

Thursday Feb 12, 15:15, room B3.02

Daan Krammer (Warwick)

The braid group of Zⁿ

Abstract: The braid group B of Zⁿ is a group that sits somewhere between braid groups and surface mapping class groups. It is to GL(n,Z) what the braid group is to the symmetric group.

I shall define B and sketch a proof that B is Garside and in particular has a solvable word problem. I describe an almost-finite presentation for B analogous to the usual presentation for the braid group.

I discuss somewhat speculative similarities between B and mapping class groups. For any statement about mapping class groups, solved or unsolved, it may be interesting to try to state and prove it for B. But whether B is useful remains to be seen.

Thursday Feb 19, 15:15, room B3.02

Tony O'Farrell (Maynooth)

Reversible biholomorphic germs

Abstract: PDF.

Thursday Feb 26, 15:15, room B3.02

Dmitri Zaitsev (Dublin)

Chern-Moser type normal forms for almost CR structures

Abstract: Chern-Moser normal form is a fundamental tool to classify real hypersurfaces in complex spaces up to local biholomorphic changes of coordinates. In this work we extend this normal form to CR structures that are not necessarily integrable. One of the main differences with the classical integrable case is the presence of the non-integrability tensor at the same order as the Levi form, making impossible a good quadric approximation - a key tool in the Chern-Moser theory.

Thursday Mar 5, 15:15, room B3.02

John Ratcliffe (Vanderbilt)

Fibered orbifolds and crystallographic groups

Abstract: In this talk, the relationship between normal subgroups of Euclidean crystallographic groups and orbifold fibrations of compact flat orbifolds will be described. Most compact flat orbifolds have nontrivial flat orbifold fibrations. For example, if M is a compact flat orbifold, whose orbifold fundamental group has an infinite abelianization, then M has two nontrivial canonical flat orbifold fibrations, whose respective fibers are orthogonal at each point of intersection. Flat orbifold fibrations are interesting because they underlie the geometry and topology of closed flat manifolds. This is joint work with my colleague Steven Tschantz.

Thursday Mar 5, 16:30, room MS.05

Anna Lenzhen (Lille)

Shape of a ball in Teichmueller space

Abstract: We show that extremal length along a Teichmueller geodesic is a quasi-convex function of time. We conclude that a ball in Teichmueller space is quasi-convex. Joint work with Kasra Rafi.

Thursday Mar 12, 15:15, room B3.02

Paolo Ghiggini (CNRS)

Classification of tight contact structures on small Seifert manifolds

Abstract: A tight contact structure on a 3-manifold is a nowhere integrable tangent plane field which satisfies an additional rigidity property relating it to the topology of the 3-manifold. The classification of tight contact structures up to isotopy is an intensely studied, but still poorly understood, problem in contact topology.

I will explain what is known, what is expected and what is unknown about the classification of tight contact structures on Seifert manifolds over S^2 with three singular fibres, and why it matters.