# MS Seminar (Mathematics - String Theory)

Speaker: Anton Khoroshkin (HSE Moscow) Cacti groups, an operadic point of view Tue, Jan 09, 2018, 13:15 - 14:45 Seminar Room A The category of representations over a quantum group $U_q(g)$ form a braided tensor category that produces an action of the (pure) braid groups on tensor products. Respectively, the category of crystals (which is a limit for q tends to zero) form a coboundary category together with an action of (pure) cacti group on tensor products. The little discs operad is an operad whose space of $n$-ary operations is the Eilenberg-Maclane space of the pure braid groups with $n$ braids. Correspondingly, the real loci of the moduli spaces of stable rational curves with marked points assemble an operad of the Eilenberg-Maclane spaces of pure cacti groups. I will present the detailed description of the latter operad as well as its deformation theory and relationships with the little discs operad. Among different applications I will prove rational K(\pi,1) property of the latter moduli spaces as well as other interesting properties of the pure cacti groups that were conjectured by P.Etingof, A.Henriques, J.Kamnitzer and E.Rains in the seminal paper arXiv:math/0507514. Talk is based on the joint work with Thomas Willwacher.