Abstract: 
I will give a final overview of the theory of P^nfunctors developed by Rina Anno and I in arXiv:1905.05740. After some formal definitions, I will concentrate on four new families of examples of non–split P^nfunctors: spherical functors, extensions by zero, cyclic covers, and family Ptwists. For the latter, I will explain the associated Mukai flop picture and the "flopflop = twist" formula. I will then discuss our proof of Segal's conjecture: any split P^nfunctor deforms to a spherical functor from a noncommutative line bundle over the base category. I will finish with some speculation: an approach to the converse of Segal's conjecture, the related question of whether the P^nfunctor data can be 'reconstructed' from its first order truncation (the Ptwist data), the reasons why Segal's conjecture doesn't generalise straightforwardly to nonsplit P^nfunctors, and the relation between all this and a work in progress with Alexei Bondal on perverse schobers on orbifolds.
