My colleague David Stewart organized a workshop on Pseudo-reductive groups in September 2017, which was partially funded by the Heilbronn Institute.
In this workshop, Gopal Prasad gave a mini-course on his work with Conrad and Gabber on pseudo-reductive groups.
I contributed a lecture on Reductive subgroups of parahoric group schemes. Here is the abstract for my talk:
Let K be the field of fractions of a complete discrete valuation ring A with residue field k, and let G be a connected reductive algebraic group over K. Suppose P is a parahoric group scheme attached to G. In particular, P is a smooth affine A-group scheme having generic fiber PK = G; the group scheme P is in general not reductive over A. Assume that G splits over an unramified extension of K.
The talk will give an overview of two results about G.
First, there is a closed and reductive A-subgroup scheme M of P for which the special fiber Mk is a Levi factor of Pk. Moreover, the reductive subgroups of G = PK of the form MK may be characterized.
Second, let X be a nilpotent section in Lie (P). We say that X is balanced if the fibers CK and Ck are smooth group schemes of the same dimension, where C = CP(X) is the scheme theoretic centralizer of X. If X0 is a given nilpotent element in the Lie algebra of the reductive quotient of the special fiber Pk, we give results on the possible lifts of X0 to a balanced nilpotent section X ∈ Lie (P).