Classically, one puts an algebraic structure on certain "congruence" quotients of the upper half plane by interpreting them as spaces parametrising elliptic curves with certain level structures on their torsion subgroups. However, the non-congruence quotients don't admit such a straightforward description.

 

We will sketch the classical theory of congruence modular curves and level structures, and then discuss a preprint by W. Chen which extends the above notions to non-congruence modular curves by considering so-called Teichmueller level structures on the fundamental groups of punctured elliptic curves.

In this talk, I will present some recent developments in the theory of pseudo-differential operators on Lie groups. First I will discuss why `reasonable' Lie groups are the interesting manifolds where one can develop global symbolic pseudo-differential calculi. I will also give a brief overview of the analysis in the context of Lie groups. I will conclude with some recent works developing pseudo-differential calculi on certain classes of Lie groups.