In this talk we will discuss a new approach to non rationality of projective varieties based on HMS. Examples will be discussed.

In this talk I will discuss a new proposal for constructing quantizations of holomorphic Poisson structures, and generalized complex manifolds more generally, which is based on using the A model of an associated symplectic manifold known as a Morita equivalence. This construction will be illustrated through the example of toric Poisson structures.

This talk is an update on joint work with Geoff Penington on extending Morse theory to smooth functions on compact manifolds with very mild nondegeneracy assumptions. The only requirement is that the critical locus should have just finitely many connected components. To such a function we associate a quiver with vertices labelled by the connected components of the critical locus. The analogue of the Morse–Witten complex in this situation is a spectral sequence of multicomplexes supported on this quiver which abuts to the homology of the manifold.