BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Kevin Ivan Piterman DTSTART:20231204T103000Z DTEND:20231204T113000Z DTSTAMP:20260422T135612Z UID:BilTop/71 DESCRIPTION:Title: Advances on Quillen's conjecture\nby Kevin Ivan Piterman as part of Bi lkent Topology Seminar\n\nLecture held in SB-Z11.\n\nAbstract\nThe study o f the p-subgroup complexes began motivated by group cohomology and equivar iant cohomology of topological spaces "modulo the prime p". For example\, Kenneth Brown proved that the reduced Euler characteristic of this complex is divisible by the size of a Sylow p-subgroup\, giving rise to a sort of "Homological Sylow theorem". Later\, he showed that the mod-p equivariant cohomology of the p-subgroup complex of a finite group coincides with the mod-p cohomology of the group. Deeper relations with finite group theory\ , representation theory\, and finite geometries were also explored. For in stance\, uniqueness of certain simple groups\, finite geometries for spora dic groups\, Lefschetz modules\, and\, more recently\, endotrivial modules .\n\nIn 1978\, Daniel Quillen conjectured that the poset of non-trivial p- subgroups of a finite group G is contractible if and only if G has non-tri vial p-core. Quillen established the conjecture for solvable groups and so me families of groups of Lie type. The major step towards the resolution o f the conjecture was done by Michael Aschbacher and Stephen D. Smith at th e beginning of the nineties. They roughly proved that if p>5 and G is a gr oup of minimal order failing the conjecture\, then G contains a simple com ponent PSU(n\,q^2) failing a certain homological condition denoted by (QD) (namely\, the top-degree homology group of its p-subgroup poset does not vanish).\n\nIn this talk\, I will present recent advances in the conjectur e\, with a particular focus on the prime p=2\, which was not covered by th e methods developed by Aschbacher-Smith. In particular\, we show that the study of the conjecture for the prime p=2 basically reduces to studying (Q D) on the poset of p-subgroups of certain families of classical groups. Pa rt of this work is in collaboration with S.D. Smith\n LOCATION:https://researchseminars.org/talk/BilTop/71/ END:VEVENT END:VCALENDAR