Daniel Erman |
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. |
---|---|

Brendan Hassett |
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. |

Elham Izadi |
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. |

Martijn Kool |
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. |

James McKernan |
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. |

Martin Olsson |
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. |