Multiplicative quiver varieties are a variant of Nakajima's "additive" quiver varieties which were introduced by Crawley-Boevey and Shaw.
They arise naturally in the study of various moduli spaces, in particular in Boalch's work on irregular connections. In this talk we will discuss joint work with Tom Nevins which shows that the tautological classes for these varieties generate the largest possible subalgebra of the cohomology ring, namely the pure part.

The Hitchin connection is a flat projective connection on bundles of non-abelian theta-functions over the moduli space of curves, originally introduced by Hitchin in a Kahler context.  We will describe a purely algebra-geometric construction of this connection that also works in (most)positive characteristics.  A key ingredient is an alternative to the Narasimhan-Atiyah-Bott Kahler form on the moduli space of bundles on a curve.  We will comment on the connection with some related topics, such as the Grothendieck-Katz p-curvature conjecture.  This is joint work with Baier, Bolognesi and Pauly.