It is well-known that one can attach Galois representations to modular forms. In the case of cusp forms, the corresponding l-adic Galois representations are irreducible for every prime l, while in the case of Eisenstein series, the corresponding Galois representations are reducible. The Langlands correspondence is expected to generalise this picture, with cuspidal automorphic representations always giving rise to irreducible Galois representations. In the cuspidal, polarized, regular algebraic setting over a CM field, a construction of Galois representations is known, but their irreducibility is still an open problem in general. I will discuss recent joint work with Zachary Feng establishing new instances of irreducibility, and outline how our methods extend some previous approaches to this problem.

Dr Anna Lisa Vari will talk about: 'The orbital structure of the Hill's problem'

Hill’s problem is a limiting case of the circular restricted gravitational three-body problem in which the mass ratio between the two massive bodies tends to zero, leaving a small region surrounding the secondary in which it remains gravitationally dominant. Originally formulated in terms of point masses, Hill’s problem may be modified to include a secondary of finite extent, thus providing a more realistic description of the dynamics internal to a stellar cluster orbiting within a host galaxy. By considering stellar energies above the cluster escape energy, we may investigate the dynamics that underpin the process of stellar escape from star clusters -- a topical issue in contemporary astrophysics. Specifically, we construct a self-consistent formulation of Hill’s problem using a tidally perturbed cluster model for the secondary body. The behaviour of energetically unbound stellar orbits within such a self-consistent problem, as characterised using Poincaré surfaces of section, is then numerically explored via a structure-preserving integrator, revealing a previously unknown bifurcation in the orbital structure.