'We study an optimal switching problem with a state constraint: the controller is only allowed to choose strategies that keep the controlled diffusion in a closed domain. We prove that the value function associated to the weak formulation of this problem is the limit of the value function associated to an unconstrained switching problem with penalized coefficients, as the penalization parameter goes to infinity. This convergence allows to set a dynamic programming principle for the constrained switching problem. We then prove that the value function is a constrained viscosity solution to a system of variational inequalities (SVI for short). We finally prove that the value function is the maximal solution to this SVI. All our results are obtained without any regularity assumption on the constraint domain.’

Let ${\cal E}$ be a family of elliptic curves over a base variety defined over $\mathbb C$. An additive extension ${\cal G}$ of ${\cal E}$ is a family of algebraic groups which fits into an exact sequence of group schemes $0\rightarrow {\mathbb G}_{\rm a}\rightarrow {\cal G}\rightarrow {\cal E}\rightarrow 0$. We can define the special subvarieties of ${\cal G}$ to be families of algebraic groups over the same base contained in ${\cal G}$. The relative Manin-Mumford conjecture suggests that the intersection of a curve in ${\cal G}$ with the special subvarieties of dimension 0 is contained in a finite union of special subvarieties.

To prove this we can assume that the family ${\cal E}$ is the Legendre family and then follow the strategy employed by Masser-Zannier for their proof of the relative Manin-Mumford conjecture for the fibred product of two legendre families. This has applications to classical problems such as the theory of elementary integration and Pell's equation in polynomials.

The classification of irreducible representations of pin double covers of Weyl groups was initiated by Schur (1911) for the symmetric group and was completed for the other groups by A. Morris, Read and others about 40 years ago. Recently, a new relation between these projective representations, graded Springer representations, and the geometry of the nilpotent cone has emerged. I will explain these connections and the relation with a Dirac operator for (extended) graded affine Hecke algebras.  The talk is partly based on joint work with Xuhua He.