BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Michele Ancona (Côte d'Azur) DTSTART:20260313T124000Z DTEND:20260313T134000Z DTSTAMP:20260423T021250Z UID:OBAGS/81 DESCRIPTION:Title: H arnack manifolds\nby Michele Ancona (Côte d'Azur) as part of ODTU-Bil kent Algebraic Geometry Seminars\n\n\nAbstract\nIn 1876\, Axel Harnack pro ved in a foundational article that\n\n1) every real algebraic curve of deg ree d in RP^2 has at most (d-1)(d-2)/2 + 1 connected components\;\n\n2) fo r every d there exists a curve of degree d with exactly this number of con nected components.\n\n\nOver the past 150 years\, these results have playe d a central role in the study of the topology of real algebraic varieties. The first part of Harnack’s theorem generalizes to the so-called Smith –Floyd inequality for arbitrary real algebraic varieties: the sum of the Betti numbers of the real part is at most the corresponding sum for the c omplex part. Despite spectacular advances\, the generalization of the seco nd part of Harnack’s theorem remains open in the case of projective hype rsurfaces.\n\nFor these\, however\, Ilia Itenberg and Oleg Viro showed tha t the Smith–Floyd inequality is asymptotically optimal by using the comb inatorial patchworking technique. In joint work with Erwan Brugallé and J ean-Yves Welschinger\, we show that an elementary generalization of Harnac k’s original construction method in dimension 2 yields this asymptotic o ptimality for any ample line bundle on a real algebraic variety. Beyond Be tti numbers\, we also describe the diffeomorphism type of an open subset o f these topologically rich varieties.\n LOCATION:https://researchseminars.org/talk/OBAGS/81/ END:VEVENT END:VCALENDAR