Title:  Finiteness for Tate-Shafarevich sets of affine algebraic groups

Abstract: A fundamental problem in the theory of abelian varieties over 
global fields is to prove finiteness of their Tate-Shafarevich groups.  
Even for elliptic curves this is known only in special cases.  But a 
"Tate-Shafarevich set" can be defined for any algebraic group over a global 
field, and it can be very useful even in the affine case, where a lot is known.  
In particular, Borel and Serre proved the finiteness for affine groups over 
number fields, and Borel and Prasad proved it for some important cases  over 
global function fields.  But the general function field case for affine groups 
was wide open, yet is also quite natural (for reasons I will explain).  
I will present a proof in general (at least away from characteristic 2), by 
using the new structure theory for pseudo-reductive groups 
(developed jointly with Gabber and G. Prasad).