BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Joachim Kock DTSTART:20210408T170000Z DTEND:20210408T180000Z DTSTAMP:20260508T135004Z UID:ToposInstituteColloquium/7 DESCRIPTION:Title: Noncrossing hyperchords and free probability\nby Joac him Kock as part of Topos Institute Colloquium\n\n\nAbstract\nFree probabi lity is a noncommutative probability theory introduced by Voiculescu in th e 1980s\, motivated by operator algebras and free groups\, and useful in r andom matrix theory. Where classical independence relates to the tensor pr oduct of algebras\, free independence relates to the free product of algeb ras. Speicher discovered the combinatorial substrate of the theory: noncro ssing partitions. He derived the free cumulant-moment relations from Möbi us inversion in the incidence algebra of the lattice of noncrossing partit ions\, and used it\, via two reduction procedures\, to model free multipli cative convolution. A crucial ingredient\, which has no analogue in the cl assical setting\, is the notion of Kreweras complement of a noncrossing pa rtition. In this talk\, after a long introduction to these topics\, I will explain some more categorical viewpoints. A first step is an operad of no ncrossing partitions. A second step is a decomposition space (2-Segal spac e) Y of noncrossing hyperchords\, whose simplicial structure encodes highe r versions of Kreweras complementation. The incidence bialgebra of Y is a direct combinatorial model for free multiplicative convolution. It is rela ted to the previous models by the standard simplicial notion of decalage: the first decalage of Y gives the (two-sided bar construction of the) oper ad\, and the second decalage gives the lattice. These two decalages encode precisely Speicher's two reductions.\n\nThis is joint work with Kurusch E brahimi-Fard\, Loïc Foissy\, and Frédéric Patras.\n LOCATION:https://researchseminars.org/talk/ToposInstituteColloquium/7/ END:VEVENT END:VCALENDAR