| Abstract: |
I will give a final overview of the theory of P^n-functors 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^n-functors: spherical functors, extensions by zero, cyclic covers, and family P-twists. For the latter, I will explain the associated Mukai flop picture and the "flop-flop = twist" formula. I will then discuss our proof of Segal's conjecture: any split P^n-functor deforms to a spherical functor from a non-commutative 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^n-functor data can be 'reconstructed' from its first order truncation (the P-twist data), the reasons why Segal's conjecture doesn't generalise straightforwardly to non-split P^n-functors, and the relation between all this and a work in progress with Alexei Bondal on perverse schobers on orbifolds.
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