I will describe a new method for constructing conformal blocks for the Virasoro vertex algebra with central charge c=1, by "nonabelianization", relating them to conformal blocks for the Heisenberg algebra on a branched double cover. The construction is joint work with Qianyu Hao. Special cases give rise to formulas for tau-functions and solutions of integrable systems of PDE, such as Painleve I and its higher analogues. The talk will be reasonably self-contained (in particular I will explain what a conformal block is).

We use Non-Reductive GIT to construct compactifications of Hilbert schemes of complete intersections. We then study ample line bundles on these compactifications in order to construct moduli spaces of complete intersections for certain degree types.

I will review recent progress on extending the Minimal Model Program to algebraically integrable foliations, focusing on applications such as the canonical bundle formula and recent results toward the boundedness of Fano foliations.