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.

A Ricci iteration is a sequence of Riemannian metrics on a manifold such that every metric in the sequence is equal to the Ricci curvature of the next metric. These sequences of metrics were introduced by Rubinstein to provide a discretisation of the Ricci flow. In this talk, I will discuss the relationship between the Ricci iteration and the Ricci flow. I will also describe a recent result concerning the existence and convergence of Ricci iterations close to certain Einstein metrics. (Joint work with Max Hallgren)