Daniel Erman Title: The structure of sheaf cohomology groups Abstract: Fix a coherent sheaf F supported on a subvariety of projective space. What can we say about the ranks of the sheaf cohomology groups of F and its twists F(i)? I will first discuss some elementary approaches to this problem, and then explain how these elementary approaches are refined by results of Eisenbud, Schreyer, myself, and others. Title: Rational curves on holomorphic symplectic varieties Abstract: Holomorphic symplectic manifolds are higher-dimensional generalizations of K3 surfaces. We will survey results on the structure of rational curves on these varieties, drawing on new ideas from stability conditions and derived categories. Title: Families of lines in pencils of cubic surfaces and Prym-Tyurin varieties. Abstract: I will discuss some properties of pencils of cubic surfaces, their degenerations, deformations and associated Prym-Tyurin varieties with an application to abelian 5-folds and 6-folds. Title: Curves on Surfaces Abstract: Curves on surfaces The Hilbert scheme of curves in class \beta on a smooth projective surface S carries a natural virtual cycle. In many cases this cycle is zero (often when S has a holomorphic 2-form and \beta is not sub-canonical). However, in these cases one can often remove part of the obstruction bundle and obtain a non-trivial reduced virtual cycle. Both cycles have interesting applications. (1) Both are related to Pandharipande-Thomas' stable pair invariants on the total space of the canonical bundle over S. (2) The reduced virtual cycle is related to Severi degrees and classical curve counting on S. (3) The non-reduced virtual cycle is related to the Seiberg-Witten invariants of S. Title: Toroidal Modifications Abstract: Shokurov has conjectured that the set of log discrepancies of singularities in a fixed dimension satsfies the ACC (ascending chain condition). As part of an attempt to approach this conjecture, we propose a conjectural partial classification of log terminal singularities, which uses toric singularities as a building block. Title: Actions of correspondences and local terms Abstract: In this talk I will give an overview of a continuing project aimed at understanding motivic aspects of correspondences acting on $\ell$-adic sheaves. This work is motivated by trying to prove various independence of $\ell$ results, but also leads to several interesting problems in intersection theory which I will highlight.