BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Igor Pak (UCLA) DTSTART:20220504T163000Z DTEND:20220504T183000Z DTSTAMP:20260423T024752Z UID:Vinberg/16 DESCRIPTION:Title: Combinatorial inequalities\nby Igor Pak (UCLA) as part of The Vinberg Lecture Series\n\n\nAbstract\nIn the ocean of combinatorial inequalities\ , two islands are especially difficult. First\, Mason's conjectures say t hat the number of forests in a graph with k edges is log-concave. More ge nerally\, the number of independent sets of size k in a matroid is log-con cave. Versions of these results were established just recently\, in a rem arkable series of papers by Huh and others\, inspired by algebro-geometric considerations. \n\nSecond\, Stanley's inequality for the numbers of lin ear extensions of a poset with value k at a given poset element\, is log-c oncave. This was originally conjectured by Chung\, Fishburn and Graham\, and famously proved by Stanley in 1981 using the Alexandrov–Fenchel ineq ualities in convex geometry. No direct combinatorial proof for either res ult is known. Why not? \n\nIn the first part of the talk we will survey a number of combinatorial inequalities. We then present a new framework o f combinatorial atlas which allows one to give elementary proofs of the tw o results above\, and extend them in several directions. This talk is aim ed at the general audience.\n LOCATION:https://researchseminars.org/talk/Vinberg/16/ END:VEVENT END:VCALENDAR