BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Matthieu Piquerez (University of Nantes) DTSTART:20230214T200000Z DTEND:20230214T210000Z DTSTAMP:20260423T040216Z UID:Matroids/36 DESCRIPTION:Title: Tropical Hodge theory\nby Matthieu Piquerez (University of Nantes) a s part of Matroids - Combinatorics\, Algebra and Geometry Seminar\n\nLectu re held in Room 210 The Fields Institute.\n\nAbstract\nDuring the last dec ade\, several important long-standing conjectures about matroids\, as the Heron-Rota-Welsh conjecture\, has been solved thanks to the development of the combinatorial Hodge theory by Huh and his collaborators. Classical Ho dge theory is about the cohomology of complex varieties. For matroids repr esentable over the complex field\, this theory applied to some complex var ieties associated to the matroids implies the Heron-Rota-Welsh conjecture. For a general matroid\, Adiprasito\, Huh and Katz achieved to develop a c ombinatorial Hodge theory for (Chow rings of) matroids which works as if o ne can associate a complex variety to the matroid\, though this is not the case. The proof is very clever but does not give much insight into why th is combinatorial Hodge theory works in general.\n\nActually\, every matroi d is in some sense representable over the tropical hyperfield. Moreover\, in a joint work with Amini\, we developed a tropical Hodge theory. Hence\, to every matroid one can associate a tropical variety (the canonically co mpactified Bergman fan)\, and the Hodge properties of this variety imply t he Heron-Rota-Welsh conjecture. We thus get a geometric proof of the conje cture\, as well as an extension of the applicability of the combinatorial Hodge theory. The heart of our proof relies on a very interesting inductio n\, based on the deletion-contraction induction."\n LOCATION:https://researchseminars.org/talk/Matroids/36/ END:VEVENT END:VCALENDAR