Wojciech Chachólski: Dynamics and symmetries of idempotent deformations of groups

The proofs of Ravenel’s conjectures and their reinterpretations in the form of the classification of the homotopy idempotent functors of spectra that commute with telescopes by Devintaz-Hopkins-Smith were a culmination of a few decades of progress achieved in stable homotopy theory. Similar classification for spaces turned out to be out of reach as it is related to the homotopy groups of spheres. To understand possible difficulties one strategy has been to ask analogous classification questions in other settings with a hope that the results might shed some light on possible difficulties in the case of topological spaces. That led to extensive research particularly in the algebraic settings among them in the category of groups. Roughly speaking an idempotent deformation is a functors of groups that applied twice gives the same functor. Such functors preserve a lot structural properties of groups: nilpotence, solvability, finiteness. It is not known however if finite presentability is preserved. In my talk I will describe the action of the idempotent deformation on finite groups showing that the orbits of finite simple groups can have at most 7 elements. One can wonder what is dynamics of such operations on broader collection of groups.