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SUMMARY:Ambitropical Convexity\, Mean Payoff Games and Nonarchimedean Conv
ex Programming (in person)
DTSTART;VALUE=DATE-TIME:20211202T135000Z
DTEND;VALUE=DATE-TIME:20211202T144000Z
DTSTAMP;VALUE=DATE-TIME:20220810T080827Z
UID:indico-contribution-5976@indico.math.cnrs.fr
DESCRIPTION:Speakers: Stéphane Gaubert (INRIA & Ecole Polytechnique)\nCon
vex sets can be defined over ordered fields with a non-archimedean valuati
on. Then\, tropical convex sets arise as images by the valuation of non-ar
chimedean convex sets. The tropicalization of polyhedra and spectrahedra c
an be described in terms of deterministic and stochastic games with mean p
ayoff\, being characterized in terms of sub or super-fixed point sets of S
hapley operators\, which determine the value of the game. This is motivate
d by open complexity issues in linear programming. We shall discuss here e
specially a generalization of tropical convexity: considering fixed point
sets of Shapley operators\, instead of sub or super-fixed points sets\, le
ads to a richer “ambitropical” theory\, which includes tropical convex
ity and its dual in a unified framework. We shall present several characte
rizations of ambitropical convex sets\, with features related to normed sp
aces (nonexpansive retracts and hyperconvexity)\, lattice theory (order pr
eserving retracts)\, or of a combinatorial nature (cell decompositions in
alcoved polyhedra).\nThe results on ambitropical convexity is from a work
with Akian and Vannucci\; the ones on the tropicalization of nonarchimedea
n convex sets are from works with Allamigeon\, Benchimol\, Joswig and Skom
ra.\n\nhttps://indico.math.cnrs.fr/event/7040/contributions/5976/
LOCATION:Le Bois-Marie Centre de conférences Marilyn et James Simons
URL:https://indico.math.cnrs.fr/event/7040/contributions/5976/
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