I will present basic ideas of a joint work in progress with M. Belolipetsky in which we try to determine the orbifolds given in the title, for odd dimensions starting from five.

In this talk I shall report recent progress on continuous-time behavioural portfolio choice under Kahneman and Tversky's (cumulative) prospect theory, featuring S-shaped utility functions and probability distortions. It is shown that the model well-posedness becomes a prominent issue in such a behavioural model. The optimal terminal wealth positions, derived in fairly explicit forms, possess surprisingly simple structure, leading to gambling strategies that bet on good states of the world while accepting fixed, known losses in case of bad ones. Time permitting I will also discuss on the incomplete market and single-period models as well as the equity premium puzzle.

The vortex sheet problem involves solving the Euler equations in two dimensions for an incompressible fluid when the initial vorticity is a measure restricted to a curve. In 1991, J.-M. Delort showed the existence of a weak solution to the vortex sheets problem under the additional assumption that the initial vorticity is signed. We will discuss a proof of this result by L.C. Evans and S. Muller which utilizes properties of the Hardy Space H^1. This is a candidacy exam.

It is observed that the group of nonzero algebraic numbers modulo roots of unity is in fact a vector space G, written multiplicatively, over the rationals with norm given by the Weil Height. How do classical notions such as the degree of an algebraic number generalize to this context? (The Q-vector space dimension of a number field Q(a)/Q is hidden in this new group.) We show the absolute Galois group on G preserves this information, generalizes the concept of the degree of an algebraic number, and leads to an equivalent formulation of Lehmer's problem in this new context. The study of this notion of degree will also be seen to generalize the results of Boyd and Ferguson which concern the circumstances under which the minimal polynomial f(x) in Z[x] of an algebraic number alpha can be reduced in the sense that f(x)=g(x^m) for some g(x) in Z[x].

We study smooth families of diffeomorphisms of that have invariant manifolds arising from a perturbation of a saddle connection. In this context, chaotic transport appears naturally. This version of phase space transport is based on structures that are formed of pieces of normally hyperbolic invariant manifolds, called lobes. We present a formula that measures the volume of such lobes and explore their topology and possible measures of their complexity.

In this talk I will show that the geometric mechanism for detecting global instability introduced in a recent paper by Delshams, de la Llave and Seara can be applied to a general class of a-priori unstable Hamiltonian systems. The proof of this results includes a precise study of the effect of the resonances present in the system, which is remarkably useful not only for the problem of diffusion but also for the comprehension of other phenomena present in physical models. I will also consider a particular example and I will show that it admits diffusing orbits that follow this geometric mechanism.