Séminaires : Séminaire d'Algèbre

Equipe(s) : gr,
Responsables :J. Alev, D. Hernandez, B. Keller, Th. Levasseur, et S. Morier-Genoud.
Email des responsables : Jacques Alev <jacques.alev@univ-reims.fr>, David Hernandez <david.hernandez@imj-prg.fr>, Bernhard Keller <bernhard.keller@imj-prg.fr>, Thierry Levasseur <Thierry.Levasseur@univ-brest.fr>, Sophie Morier-Genoud <sophie.morier-genoud@imj-prg.fr>
Salle : 001
Adresse :IHP
Description
 

 


Orateur(s) Luca MOCI - Paris,
Titre Matroids over a ring: motivations, examples, applications.
Date14/10/2013
Horaire14:00 à 15:00
RésumeSeveral objects can be associated to a list of vectors with integer coordinates: among others, a family of tori called toric arrangement, a convex polytope called zonotope, a function called vector partition function. These objects and their relations have been described in a recent book by De Concini and Procesi. The linear algebra of the list of vectors is retained by the combinatorial notion of a matroid; but several properties of the objects above depend also on the arithmetics of the list. This can be encoded by the notion of a ``matroid over Z''. Similarly, applications to tropical geometry suggest the introduction of matroids over a discrete valuation ring. Motivated by the examples above, we introduce the more general notion of a ``matroid over a commutative ring R''. Such a matroid arises for example from a list of elements in a R-module. When R is a Dedekind domain, we can extend the usual properties and operations holding for matroids (e.g., duality). We can also compute the Tutte-Grothendieck ring of matroids over R; the class of a matroid in such a ring specializes to several invariants, such as the Tutte polynomial and the Tutte quasipolynomial. We will also outline other possible applications and open problems. (Joint work with Alex Fink).
Salle001
AdresseIHP
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