Manifolds are important entities in dynamical systems, and organise space into regions in which different motions occur. For example, intersections between stable and unstable manifolds in discrete systems result in chaotic motion. This talk will focus on manifolds and their locations in continuous dynamical systems, and in particular on Melnikov's method and its adaptations for determining the effect of perturbations on manifolds. The relevance of such adaptations to a surprising range of applications will be shown, in addition to recent theoretical developments inspired by such problems. The applications addressed in this talk include understanding the motion of fluid near oceanic eddies and currents, optimising mixing in nano-fluidic devices in order to improve reactions, computing the speed of a flame front, and finding the spreading rate of bacterial colonies.

The problem of equilibrium pricing in dynamically incomplete financial markets is one of the oldest problems in mathematical economics. The problem of equilibrium pricing is well understood for the benchmark case of complete markets where all risk factors can be hedged using the available assets. When markets are incomplete the situation is more involved, and to date no unified approach to incomplete markets is available. In this talk we review some recent results on equilibrium pricing in incomplete markets in discrete time when the market participants evaluate their risk exposures using dynamic risk measures. For such market situations we establish existence and uniqueness of equilibrium results and show that the problem of dynamic equilibrium pricing can be reduced to a recursive sequence of static one-period problems. When the flow of information is generated by independent random walks the equilibrium dynamics can be described by a coupled system of backward stochastic difference equations which renders our approach easily amenable to numerical simulations. We also comment on some of the mathematical challenges that currently prevent us from translating our results from discrete to continuous time, including the lack of existence and differentiability of solutions results for coupled systems of backward stochastic differential equations with non Lipschitz continuous drivers. The talk is based on joint work with Patrick Cheridito (Princeton University), Michael Kupper (Humboldt University) and Traian A. Pirvu (McMaster University)