Formal geometry of Lie groupoids
I will give a brief survey of the formal geometry of groupoids. I will recall the definition of Lie groupoid, provide a few examples, construct the associated Lie algebroid, its universal enveloping algebra, and its dual: the space of jets. Next, I will explain how space of jets inherits a structure of Hopf algebroid. Finally, I will describe the Hochschild cohomology and homology of this Hopf algebroid and endow them with a noncommutative calculus structure.
Introduction to Differentiable Stacks
In this mini-course, I will provide a brief introduction to the theory of differentiable stacks in terms of Lie groupoids. I will introduce (+1)-shifted Poisson and symplectic structures on differentiable stacks and discuss their applications in symplectic geometry and momentum map theory.
Categorical quantization of Kaehler manifolds
Generalizing deformation quantizations with separation of variables of a Kaehler manifold M, I will introduce a construction of an enriched category DQ as a quantization of the category of Hermitian holomorphic vector bundles over M with morphisms being smooth sections of hom-bundles.
I will then define quantizable morphisms among objects in DQ as a generalization of Chan- Leung-Li’s notion of quantizable functions. Upon evaluation of quantizable morphisms at h = 1/k, an enriched category DQqu,k is obtained. We show that, when M is prequantizable, DQqu,k is equivalent to the category GQ of holomorphic vector bundles over M with morphisms being holomorphic differential operators, via a functor obtained from Bargmann-Fock actions. This generalizes Chan-Leung-Li’s result that the sheaf of level-k quantizable functions is isomorphic to the sheaf of holomorphic sections of kth tensor power of a prequantum line bundle of M.