Abstract: The notion of a Lie conformal algebra (LCA) comes from physics, and is related to the operator product expansion. An LCA is a module over a ring of differential operators with constant coefficients, and with a bracket which may be seen as a deformation of a Lie bracket. LCA are related to linearly compact differential Lie algebras via the so-called annihilation functor. Using this observation and Cartan's classification of linearly compact simple Lie algebras, Bakalov, D'Andrea and Kac classified finite simple LCA in 2000.

I will define the notion of LCA over a ring $R$ of differential operators with not necessarily constant coefficients, extending the known one for $R=K[x]$. I will explain why it is natural to study such an object and will suggest an approach for the classification of finite simple LCA over arbitrary differential fields.

Abstract: This will be a survey talk about connections between the topology of a manifold and its group of symmetries. I will illustrate this theme by discussing finite group actions on spheres and products of spheres, and infinite discrete groups acting properly discontinuously on products of spheres and Euclidean spaces.