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Documents  Dickmann, Max | enregistrements trouvés : 8

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Research talks;Algebra;Number Theory

We give a survey on recent advances in Grothendiek's program of anabelian geometry to characterize arithmetic and geometric objects in Galois theoretic terms. Valuation theory plays a key role in these developments, thus confirming its well deserved place in mainstream mathematics.
The talk notes are available in the PDF file at the bottom of the page.

12F10 ; 12J10 ; 12L12

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Publications mathematiques de l'universite paris vii , 0033

Localisation : Publication 1er étage

structures algebriques ordonnes # structures ordonnees

06-06 ; 06Fxx

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- xviii; 366 p.
ISBN 978-1-4704-2966-9

Contemporary mathematics , 0697

Localisation : Collection 1er étage

géométrie algébrique # structure algébrique ordonnée # théorie des modèles # forme quadratique # semigroupe

03Cxx ; 06Fxx ; 11Exx ; 12-XX ; 14Pxx ; 14Qxx ; 32Sxx ; 44A60 ; 54C30 ; 58A07

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Research talks;Exposés de recherche;Algebra;Logic and Foundations

The field of Laurent series (with real coefficients, say) has a natural derivation but is too small to be closed under integration and other natural operations such as taking logarithms of positive elements. The field has a natural extension to a field of generalized series, the ordered differential field of transseries, where these defects are remedied in a radical way. I will sketch this field of transseries. Recently it was established (Aschenbrenner, Van der Hoeven, vdD) that the differential field of transseries also has very good model theoretic properties. I hope to discuss this in the later part of my talk. The field of Laurent series (with real coefficients, say) has a natural derivation but is too small to be closed under integration and other natural operations such as taking logarithms of positive elements. The field has a natural extension to a field of generalized series, the ordered differential field of transseries, where these defects are remedied in a radical way. I will sketch this field of transseries. Recently it was established ...

12L12 ; 12H05 ; 03C60 ; 03C64

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Research talks;Algebra

We say that a real closed field is an IPA-real closed field if it admits an integer part (IP) which is a model of Peano Arithmetic (PA). In [2] we prove that the value group of an IPA-real closed field must satisfy very restrictive conditions (i.e. must be an exponential group in the residue field, in the sense of [4]). Combined with the main result of [1] on recursively saturated real closed fields, we obtain a valuation theoretic characterization of countable IPA-real closed fields. Expanding on [3], we conclude the talk by considering recursively saturated o-minimal expansions of real closed fields and their IPs. We say that a real closed field is an IPA-real closed field if it admits an integer part (IP) which is a model of Peano Arithmetic (PA). In [2] we prove that the value group of an IPA-real closed field must satisfy very restrictive conditions (i.e. must be an exponential group in the residue field, in the sense of [4]). Combined with the main result of [1] on recursively saturated real closed fields, we obtain a valuation theoretic char...

06A05 ; 12J10 ; 12J15 ; 12L12 ; 13A18

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Research talks;Algebra;Algebraic and Complex Geometry

We will describe a new approach to the study of discrete dynamical systems on the real line, which consists in considering their orbits as "fractal objetcs". In particular, the formal classification of analytic systems can be reproven with this method. We will also explain the main lines of a program devoted to the study of some non analytic systems. These are generated by maps which admit a specific type of transseries (Dulac’s transseries) as asymptotic expansions. We will describe a new approach to the study of discrete dynamical systems on the real line, which consists in considering their orbits as "fractal objetcs". In particular, the formal classification of analytic systems can be reproven with this method. We will also explain the main lines of a program devoted to the study of some non analytic systems. These are generated by maps which admit a specific type of transseries (Dulac’s transseries) as ...

32C05 ; 14P15

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Research talks;Algebra;Logic and Foundations

O-minimalism is the first-order theory of o-minimal structures, an important class of models of which are the ultraproducts of o-minimal structures. A complete axiomatization of o-minimalism is not known, but many results are already provable in the weaker theory DCTC given by definable completeness and type completeness (a small extension of local o-minimality). In DCTC, we can already prove how many results from o-minimality (dimension theory, monotonicity, Hardy structures) carry over to this larger setting upon replacing ‘finite’ by ‘discrete, closed and bounded’. However, even then cell decomposition might fail, giving rise to a related notion of tame structures. Some new invariants also come into play: the Grothendieck ring is no longer trivial and the definable, discrete subsets form a totally ordered structure induced by an ultraproduct version of the Euler characteristic. To develop this theory, we also need another first-order property, the Discrete Pigeonhole Principle, which I cannot yet prove from DCTC. Using this, we can formulate a criterion for when an ultraproduct of o-minimal structures is again o-minimal. O-minimalism is the first-order theory of o-minimal structures, an important class of models of which are the ultraproducts of o-minimal structures. A complete axiomatization of o-minimalism is not known, but many results are already provable in the weaker theory DCTC given by definable completeness and type completeness (a small extension of local o-minimality). In DCTC, we can already prove how many results from o-minimality (dimension theory, ...

03C64

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- xi; 129 p.
ISBN 978-1-4704-1468-9

Memoirs of the american mathematical society , 1128

Localisation : Collection 1er étage

forme quadratique # anneau commutatif # anneau pré-ordonné # groupes spéciaux # fonction continue à valeurs réelles # archimédien avec inversion bornée # K-théorie des anneaux # spectre d'anneaux

11E81 ; 11E70 ; 12D15 ; 03C65 ; 06E99 ; 46E25 ; 54C40

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