I will begin by speaking about Ito SDEs on manifolds, how their meaning depends on the choice of a connection, and an example in which the Ito formulation is preferable to the more common Stratonovich one. SDEs are naturally generalised to the case of more irregular driving signals by rough differential equations (RDEs), i.e. equations driven by rough paths. I will explain how it is possible to give a coordinate-invariant definition of rough integral and rough differential equation on a manifold, even in the case of arbitrarily low regularity and when the rough path is not geometric, i.e. it does not satisfy a classical integration by parts rule. If time permits, I will end on a more recent algebraic result that makes it possible to canonically convert non-geometric RDEs to geometric ones.

In this talk, I will firstly give asymptotic formulas for the moments of the n-th derivative of the characteristic polynomials from the CUE. Secondly, I will talk about the connections between them and a solution of certain Painleve differential equation. This is joint work with Jonathan P. Keating.
 

This will be a continuation of the talk from last week (9 May).