A familiar technique in PDE theory is to use mollification to adjust a function controlled in some weak norm into a smooth function with corresponding control on its $C^k$ norm. It would be extremely useful to be able to do the same sort of regularisation for Riemannian metrics, and one might hope to use Ricci flow to do this. However, attempting to do so throws up some fundamental problems concerning the well-posedness of Ricci flow. I will explain some recent developments that allow us to use Ricci flow in this way in certain important cases. In particular, the Ricci flow will now allow us to adjust a `noncollapsed’ 3-manifold with a lower bound on its Ricci curvature through a family of such manifolds, without disturbing the Riemannian distance function too much, and so that we instantly obtain uniform bounds on the full curvature tensor and all its derivatives. These ideas lead to the resolution of some long-standing open problems in geometry.

No previous knowledge of Ricci flow will be assumed, and differential geometry prerequisites will be kept to a minimum.

Joint work with Miles Simon.
 

I'll discuss recent joint work with Arend Bayer and Ziyu Zhang in which we define a nef divisor class on moduli spaces of Bridgeland-stable objects in the derived category of coherent sheaves with compact support, generalising earlier work of Bayer and Macri for smooth projective varieties. This work forms part of a programme to study the birational geometry of moduli spaces of Bridgeland-stable objects in the derived category of varieties that need not be smooth and projective.

Suppose that we have a finite quotient singularity $\mathbb C^n/G$ admitting a crepant resolution $Y$ (i.e. a resolution with $c_1 = 0$). The cohomological McKay correspondence says that the cohomology of $Y$ has a basis given by irreducible representations of $G$ (or conjugacy classes of $G$). Such a result was proven by Batyrev when the coefficient field $\mathbb F$ of the cohomology group is $\mathbb Q$. We give an alternative proof of the cohomological McKay correspondence in some cases by computing symplectic cohomology+ of $Y$ in two different ways. This proof also extends the result to all fields $\mathbb F$ whose characteristic does not divide $|G|$ and it gives us the corresponding basis of conjugacy classes in $H^*(Y)$. We conjecture that there is an extension to certain non-crepant resolutions. This is joint work with Alex Ritter.