Benjamin Antieau (UW)
Topology and purity for torsors
I will explain how the $p$-local homotopy theory of the classifying space of the compact Lie group $PU_n$ can be used to solve a long-standing problem in algebraic geometry, namely the problem of when an unramified division algebra extends from a function field to a regular noetherian complex scheme.

Atoshi Chowdhury (Berkeley)
Stability of line bundles on reducible surfaces
Suppose X is a family of algebraic varieties with smooth generic fiber and reducible central fiber. Then the relative Picard stack of X (the moduli space of line bundles on X) is naturally non-separated: line bundles on the generic fiber will have infinitely many limits over the central fiber. I'll discuss a stability condition that can be imposed on line bundles in an effort to eliminate this non-separatedness. In particular, I'll give some results pointing to (weak) separatedness of the moduli space of semistable line bundles when X is a degeneration of algebraic surfaces.

Renzo Cavalieri (Colorado State)
Crepant Resolutions and Open Strings
Open GW invariants are virtual counts of maps from Riemann Surfaces with boundary to a symplectic manifold, with the image of the boundary constrained to lie in a Lagrangian submanifold of X. We propose a point of view on open GW invariants that encodes all information about such objects in term of sections of Givental's symplectic vector space. We make a general Crepant Resolution Statment - a comparison among the open theories of an orbifold and an appropriate resolution of its coarse space, and deduce from such a statement comparisons of more "classical" generating functions for open invariants. Our point of view shows that the open theory is well in tune with global mirror symmetry and with Iritani's integral structures - allowing us to verify our statements, to all genera and arbitrary number of boundary components (including the closed genus g CRC) for the family of threefold A_n singularities. This is joint work with Andrea Brini and Dusty Ross.