Positive solutions of nonlinear parabolic problems can have very complex behavior. However, assuming certain symmetry  conditions, it is possible to prove that the solutions converge to the space of symmetric functions. We show that this property is `stable'. More specifically if the symmetry conditions are replaced by asymptotically symmetric ones, the solutions still approach the space of symmetric functions. We discuss problems on bounded and unbounded domains and, by possibly suprising counterexamples, we show optimality of our  assumptions. As an application, we formulate new results on convergence of solutions to a single equilibrium.

This will be a colloquium-style talk giving a survey of certain classical results concerning the Galois structure of rings of integers. Starting with an elementary result in Galois theory (the normal basis theorem), one is led to a remarkable connection between the Galois module structure of rings on integers on the one hand, and the behaviour of certain analytical objects called Artin L-functions. I shall discuss this connection and how it fits into a much broader picture than one might have at first suspected.

Quasi-morphisms on the group of Hamiltonian diffeomorphisms are a convenient way to package and see various rigidity phenomenon in symplectic topology. Their general construction due to Entov and Polterovich uses Hamiltonian Floer homology and requires a certain computation in the quantum homology ring. In this talk I will explain how symplectic reduction provides a sort of geometric functoriality that allows quasi-morphisms to descend along symplectic reductions without further quantum homology computations. The proof involves a lemma about pulling quasi-morphisms back along `quasi-homomorphisms. As an application, I will present a family of closed symplectic manifolds whose group of Hamiltonian diffeomorphisms have infinite dimensional spaces of quasi-morphisms.

Both the arc operad and planar algebras are built from graphs on surfaces. Although similar in nature they were constructed with different aspects of geometry and applications in mind. With hindsight there is a striking similarity in one particular application. To elucidate this possible connection, we will proceed by introducing the arc operad and its discretization. The latter gives rise to actions on the Hochschild complex of a (Frobenius) algebra. These operations include Deligne's conjecture, its cyclic version and string topology. This discretization is also what is closely related to planar diagrams as we shall discuss.