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SUMMARY:Soutenance de thèse de Clementine Lemarie-Rieusset
DTSTART:20230915T120000Z
DTEND:20230915T150000Z
DTSTAMP:20260610T221600Z
UID:indico-event-10436@indico.math.cnrs.fr
DESCRIPTION:Titre de la thèse: Motivic knot theory\nRésumé: \nWe intro
 duce a counterpart in algebraic geometry to knot theory. Since this new th
 eory uses motivic homotopy theory (specifically\, quadratic intersection t
 heory)\, we name it motivic knot theory. We focus on motivic linking\, whi
 ch means that we study how two disjoint closed F-subschemes of an ambient 
 F-scheme can be intertwined\, i.e. linked together (where F is a perfect f
 ield). In knot theory\, the linking number of an oriented link with two co
 mponents (i.e. of two disjoint oriented knots) is an integer which counts 
 how many times one of the components turns around the other component. We 
 define counterparts in algebraic geometry to oriented links with two compo
 nents and to the linking number\; we call these latter counterparts quadra
 tic linking degrees. Our quadratic linking degrees are not necessarily int
 egers\; the ones we study the most take values in the Witt group of the gr
 ound field F\, which is a group of equivalence classes of symmetric biline
 ar forms over F (or equivalently\, of quadratic forms over F\, when the ch
 aracteristic of F is different from 2). After answering questions which na
 turally arise from these quadratic linking degrees\, we devise methods to 
 compute them. These methods rely on explicit formulas for the residue morp
 hisms of Milnor-Witt K-theory (from which boundary maps for the Rost-Schmi
 d complexes are constructed) and for the intersection product of the Rost-
 Schmid ring (and in particular of the Chow-Witt ring). Using these methods
 \, we explicitly compute our quadratic linking degrees on examples. Some o
 f these examples are inspired by knot theory\, specifically by torus links
  (including the Hopf and Solomon links).\n \n\nhttps://indico.math.cnrs.f
 r/event/10436/
LOCATION:Salle René Baire (IMB)
URL:https://indico.math.cnrs.fr/event/10436/
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