( Joshua talk): The theory of Brauer groups has a long and rich history. In recent years there seems to be a renewed interest in this area, and there have been several important developments in the last 10-15 years using sophisticated techniques. It is also one of the few topics that can be studied both from a complex algebraic geometry point of view as well as using algebraic tools, notably ́etale cohomology. In this talk we show that the \ell^n -torsion part of the cohomological Brauer groups of certain schemes associated to symmetric powers of a projective smooth curve over a separably closed field k are isomorphic, when \ell is invertible in k. The schemes considered are the symmetric powers themselves, then the corresponding Picard schemes and also certain Quot-schemes. We also obtain similar results for Prym varieties associated to certain finite covers of such curves. This is joint work with Jaya Iyer.

(Paranjape Talk): This talk is based on joint work with Madhav Nori which arose in the study of the Hodge conjecture for some K3 surfaces with complex multiplication. This leads to the study of the Generalised Hodge Conjecture on certain 3-folds. The conjecture can be resolved in some (restricted) cases.

(Sasha Talk): We will discuss some old and recent results concerning definability and decidability over subrings and algebraic extensions of Q. In particular, we will address the results concerning the big subrings of number fields and so-called p-bounded extensions of Q.

(Voight Talk): We compute an equation for an abelian surface that has everywhere good reduction over the quadratic field of discriminant 61. This surface does not admit a principal polarization, so we realize it as a factor of a Jacobian of an explicit curve of genus 3. This is joint work with Nicolas Mascot and Jeroen Sijsling.