Abstract: The famous Aharonov–Bohm effect demonstrates the non-trivial quantum mechanical effect of (singular) magnetic potentials. Although the single pole case has been studied widely using rotational and scaling invariance, the technique fails to work for the Aharonov–Bohm Hamiltonian with multiple poles. In this talk, we will discuss a new framework to study the resolvent, resonances, and wave asymptotics of the Aharonov–Bohm Hamiltonian with multiple poles. This results in an interesting behavior where scattering and wave asymptotics depend on the magnetic fluxes, interpolating the odd and even dimensional Euclidean scattering. This talk includes joint works with Tanya Christiansen and Kiril Datchev. 

Abstract: Gromov–Witten invariants enumerate curves in a variety X via stable maps. In practise, degenerate contributions lead to substantial overcounting which makes these invariants far from being optimal. When X is a Calabi–Yau threefold, a set of more fundamental curve counting invariants is provided by Gopakumar–Vafa invariants. In this talk, I will propose a refined correspondence conjecture between Gromov–Witten and Gopakumar–Vafa invariants when X carries a non-trivial torus action. This refinement mathematically realises and generalises features which were expected from the so called refined topological string in physics literature. I will present evidence for the conjecture in case X is the local projective plane. This is based on joint work with Andrea Brini.