Analytic methods in deformation quantization

Deformation quantization is a mathematical incarnation of the "inverse problem" in quantum mechanics: how can one associate to a classical system a quantum mechanical one which has the given classical system as its large-scale limit? Because of the richer structure in quantum mechanics, one does not normally expect this problem to have a single solution, and it is thus natural to consider the collection of all possible quantizations as a family, united by their common classical limit.
Though prompted by physics, this "quantization problem" has produced fascinating questions and discoveries in pure mathematics: Quantum systems naturally give rise to noncommutative algebras (as an observer cannot interact with the system without effecting it) whereas classical systems are commutative and hence can naturally be viewed as functions on a geometric space. If one can realize a noncommutative algebra in mathematics as quantum system, then computing its classical limit yields a geometric object which is likely to still reflect many important properties of the original system. It thus produces a bridge between algebra and geometry which has proved to be very illuminating.
Kontsevich's famous work in the 1990s showed that every Poisson structure (I.e mathematical object associated to classical mechanical system) has a quantization and indeed how to understand the family of possible quantizations. This led to an explosion of activity in the area, and while Kontsevich worked in the context of differentiable geometry, the appropriate analogues of his results in algebraic geometry are now understood. This has led to a beautiful recasting of many aspects of geometric representation theory: the celebrated Beilinson-Bernstein localization theorem can be viewed as a statement about the quantization of the cotangent bundle of a flag variety, a symplectic variety which has the property that it is a resolution of singularities of its affinization. Placed in this setting, a theorem which had previously seemed a very unusual property of flag varieties became the motivation for an exciting new area of "symplectic representation theory".

All the developments above have focused on "formal" quantizations: deformations where the deformation parameter is formal variable. One can also, however, consider convergence questions, that is, study analytic properties of quantizations as well as formal ones. In the related setting of filtered quantizations, this has been studied via the theory of analytic microlocalization, at least in the case of cotangent bundles of complex manifolds. This project however would seek to study analytic quantization in a number of new contexts: for example, it would already be interesting to develop the theory in the context of an arbitrary symplectic (or more ambitiously Poisson) variety, However, it would equally be natural to investigate analytic techniques in the non-Archimedean setting, that is, in the context of rigid analytic geometry. Here there is a richer array of possibilities in how one can impose convergence requirements. Recent work of Ardakov and others on rigid analytic D-modules, which can be thought of as a kind of filtered quantisation of cotangent bundles as mentioned above, gives evidence that this is a fruitful area of research. One natural goal in seeking to understand these results in the more general context of deformation quantisation would be to understand the localisations of finite W-algebras in the rigid analytic setting, following the work of Dodd-Kremnitzer in the formal setting.
This work will fit into EPSRCs areas of interest in Algebra, Geometry and Number theory.