The Zilber—Pink conjecture describes the points on an algebraic variety which have 'special' properties. In this talk, I will discuss some new results which can be proved about this, focusing on the examples of subvarieties of a torus, an abelian variety, or a product of modular curves. The method of proof is a generalisation of the Buium—Coleman proof of the Manin—Mumford conjecture. Parts of this are joint work with Sudip Pandit (KCL) and with Arnab Saha (IIT Gandhinagar).

Weighted flag varieties are generalizations of flag varieties and weighted projective spaces.  Although they are not usually homogeneous varieties, they are orbifolds and admit a torus action with isolated fixed points, and like ordinary flag varieties, their equivariant cohomology admits a Schubert basis.  This talk will be an introduction to weighted flag varieties, and will also discuss positivity.  Abe and Matsumura proved that the equivariant cohomology of weighted Grassmannians has a positivity property analogous to that for ordinary (non-weighted) flag varieties.  We prove a strengthened version of this result for arbitrary weighted flag varieties, along the way providing a geometric interpretation of the weighted roots of Abe and Matsumura.  This is joint work with Scott Larson.

In this talk, I present an explicit formula for the ADM mass of asymptotically locally Euclidean (ALE) almost Kähler manifolds. The formula expresses the mass in terms of the total Hermitian scalar curvature and topological data associated with the underlying almost complex structure, extending a result of Hein and LeBrun in the Kähler ALE case. The proof is based on a spin-c adaptation of Witten's proof of the positive mass conjecture in the spin case and is therefore distinct from previous complex-geometric methods. In dimension 4, I show that one can prove a positive mass theorem and a Penrose-type inequality for asymptotically Euclidean (AE) almost Kähler manifolds using this formula.