Title: Canonical representations of surface groups.Abstract: For $Sigma_{g,n}$ a genus $g$ surface with $n$ punctures, we study the character variety parameterizing representations of $pi_1(Sigma_{g,n})$. This character variety has a natural action of the fundamental group of the moduli space of curves. In joint work with Josh Lam and Daniel Litt, we aim to describe the points with […]

Title: On the cone conjecture for log Calabi-Yau threefolds.Abstract: Let $Y$ be a smooth projective threefold admitting a $K3$ fibration $f: Y rightarrow mathbb{P}^{1}$ with $-K_{Y} = f^{ast} mathcal{O}(1)$. Then the extremal rays of the cone of curves of $Y$ in the region $K_{Y} < 0$ are of two types: the blowup of a smooth curve (Type […]

Title: Symmetries of Fano varieties.Abstract: A landmark result of Birkar, Prokhorov, and Shramov shows that automorphism groups of Fano (or more generally rationally connected) varieties over C of a fixed dimension are uniformly Jordan. This means in particular that there is some upper bound on the size of symmetric groups acting faithfully on rationally connected varieties of […]