L3

By a "coarse median" we mean a ternary operation on a path metric space, satisfying certain conditions which generalise those of a median algebra. It can be interpreted as a kind of non-positive curvature condition, and is applicable, for example to finitely generated groups. It is a consequence of work of Behrstock and Minsky, for example, that the mapping class group of a surface satisfies this condition. We aim to give some examples, results and applications concerning this notion.

Balloons are two-dimensional spheres. Hoops are one dimensional loops. Knotted Balloons and Hoops (KBH) in 4-space behave much like the first and second fundamental groups of a topological space - hoops can be composed like in π1, balloons like in π2, and hoops "act" on balloons as π1 acts on π2. We will observe that ordinary knots and tangles in 3-space map into KBH in 4-space and become amalgams of both balloons and hoops.

We give an ansatz for a tree and wheel (that is, free-Lie and cyclic word) -valued invariant ζ of KBHs in terms of the said compositions and action and we explain its relationship with finite type invariants. We speculate that ζ is a complete evaluation of the BF topological quantum field theory in 4D, though we are not sure what that means. We show that a certain "reduction and repackaging" of ζ is an "ultimate Alexander invariant" that contains the Alexander polynomial (multivariable, if you wish), has extremely good composition properties, is evaluated in a topologically meaningful way, and is least-wasteful in a computational sense. If you believe in categorification, that's a wonderful playground.

For further information see http://www.math.toronto.edu/~drorbn/Talks/Oxford-130121/

Complex dynamical systems have been very well studied in recent years, in particular since computer graphics now enable us to peer deep into structures such as the Mandlebrot set and Julia sets, which beautifully illustrate the intricate dynamical behaviour of these systems. Using new techniques from Symbolic Dynamics, we demonstrate previously unknown properties of a class of quadratic maps on their Julia sets.