L3

In this talk, we introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric commutator subgroup of the normal closures of the meridians and give the invariants of the links obtained in this way. Moreover all homotopy groups of any dimensional spheres can be obtained from the geometric Massey products on certain links.

Bourdon's building is a negatively curved 2-complex built out of hyperbolic right-angled polygons. Its automorphism group is large (uncountable) and remarkably rich. We study, and mostly answer, the question of when there is a discrete subgroup of the automorphism group such that the quotient is a closed surface of genus g. This involves some fun elementary combinatorics, but quickly leads to open questions in group theory and number theory. This is joint work with David Futer.

"We investigate a construction of a Skorokhod embedding due to Root (1969), which has been the subject of recent interest for applications in Mathematical Finance (Dupire, Carr & Lee), where the construction has applications for model-free pricing and hedging of variance derivatives. In this context, there are two related questions: firstly of the construction of the stopping time, which is related to a free boundary problem, and in this direction, we expand on work of Dupire and Carr & Lee; secondly of the optimality of the construction, which is originally due to Rost (1976). In the financial context, optimality is connected to the construction of hedging strategies, and by giving a novel proof of the optimality of the Root construction, we are able to identify model-free hedging strategies for variance derivatives. Finally, we will present some evidence on the numerical performance of such hedges. (Joint work with Jiajie Wang)"