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TZID:Europe/Paris
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DTSTART:20210328T010000
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DTSTART;TZID=Europe/Paris:20210115T093000
DTEND;TZID=Europe/Paris:20210115T105000
DTSTAMP:20260407T143232
CREATED:20210115T083000Z
LAST-MODIFIED:20211025T103924Z
UID:8566-1610703000-1610707800@www.math.ens.psl.eu
SUMMARY:An application of surreal numbers to the asymptotic analysis of certain exponential functions
DESCRIPTION:Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant 1\, the identity function x\, and such that whenever f and g are in the set\, f+g\, fg and f^g are also in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below 2^(2^x). They did so by studying the possible limits at infinity of the quotient f(x)/g(x) of two functions in the fragment: if g is kept fixed and f varies\, the possible limits form a discrete set of real numbers of order type omega.Using the surreal numbers\, we extend the latter result to the whole class of Skolem functions and we discuss some additional progress towards the conjecture of Skolem. This is joint work with Marcello Mamino (http://arxiv.org/abs/1911.07576\, to appear in the JSL).
URL:https://www.math.ens.psl.eu/evenement/an-application-of-surreal-numbers-to-the-asymptotic-analysis-of-certain-exponential-functions/
LOCATION:Zoom
CATEGORIES:Séminaire Géométrie et théorie des modèles
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BEGIN:VEVENT
DTSTART;TZID=Europe/Paris:20210115T111000
DTEND;TZID=Europe/Paris:20210115T123000
DTSTAMP:20260407T143232
CREATED:20210115T101000Z
LAST-MODIFIED:20211025T103926Z
UID:8567-1610709000-1610713800@www.math.ens.psl.eu
SUMMARY:Solving equations in finite groups and complete amalgamation
DESCRIPTION:Roth’s theorem on arithmetic progression states that a subset A of the natural numbers of positive upper density contains an arithmetic progression of length 3\, that is\, the equation x+z=2y has a solution in A.Finitary versions of Roth’s theorem study subsets A of {0\, … \, N}\, and ask whether the same holds for sufficiently large N\, for a fixed lower bound on the density. In a similar way\, concerning finite groups\, one may study whether or not sufficiently large sets of a finite group contain solutions of an equation\, or even a system of equations. For instance\, for the equation xy=z\, Gowers (2008) showed that any subset of a finite simple non-abelian group will contain many solutions to this equation\, provided it has sufficiently large density.We will report on recent work with Amador Martin-Pizarro on how to find solutions to the above equations in the context of pseudo-finite groups\, using techniques from model theory which resonate with (a group version of) the independence theorem in simple theories due to Pillay\, Scanlon and Wagner. In this talk\, we will not discuss the technical aspects of the proof\, but present the main ideas to a general audience.
URL:https://www.math.ens.psl.eu/evenement/solving-equations-in-finite-groups-and-complete-amalgamation/
LOCATION:Zoom
CATEGORIES:Séminaire Géométrie et théorie des modèles
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