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Galois theory of parameterized differential equations and linear differential algebraic groups
 DIFFERENTIAL EQUATIONS AND QUANTUM GROUPS. IRMA LECTURES IN MATHEMATICS AND THEORETICAL PHYSICS, VOL 9
, 2006
"... We present a Galois theory of parameterized linear differential equations where the Galois groups are linear differential algebraic groups, that is, groups of matrices whose entries are functions of the parameters and satisfy a set of differential equations with respect to these parameters. We prese ..."
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We present a Galois theory of parameterized linear differential equations where the Galois groups are linear differential algebraic groups, that is, groups of matrices whose entries are functions of the parameters and satisfy a set of differential equations with respect to these parameters. We present the basic constructions and results, give examples, discuss how isomonodromic families fit into this theory and show how results from the theory of linear differential algebraic groups may be used to classify systems of second order linear differential equations.
MODEL THEORY OF DIFFERENTIALLY CLOSED FIELDS WITH SEVERAL COMMUTING DERIVATIONS
, 2007
"... In this thesis we deal with the model theory of differentially closed fields of characteristic zero with several commuting derivations. The questions we consider belong to the area of geometric stability theory. First we observe that the only known lower bound for the Lascar rank of types in differe ..."
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In this thesis we deal with the model theory of differentially closed fields of characteristic zero with several commuting derivations. The questions we consider belong to the area of geometric stability theory. First we observe that the only known lower bound for the Lascar rank of types in differentially closed fields, announced in a paper of McGrail, is false. This gives us a new class of regular types. Then we show that the generic type of the heat variety, which is one of these new types, is locally modular. So, unlike the case of ordinary differential fields, the additive group of a partial differential field has locally modular subgroups. We also classify the subgroups of the additive group of Lascar rank omega with differentialtype 1 which are nonorthogonal to fields.
RELATIVE GEOMETRIC CONFIGURATIONS
"... Abstract. This is a survey of a recent work done by the three authors, in which an analysis of geometric properties of a structure relative to a reduct is initiated. In particular, definable groups and fields in this context are considered. In a relatively 1based theory every group is definably iso ..."
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Abstract. This is a survey of a recent work done by the three authors, in which an analysis of geometric properties of a structure relative to a reduct is initiated. In particular, definable groups and fields in this context are considered. In a relatively 1based theory every group is definably isogenous to a subgroup of a group definable in the reduct. For relatively CMtrivial theories (which encompass certain Hrushovski’s amalgams, such as the fusion of two strongly minimal theories or coloured fields), we prove that every group can be mapped by a homomorphism with central kernel to a group definable in the reduct. 1.
Some notions of Dalgebraic geometry
, 2008
"... When studying typedefinable sets in closed Hasse fields, we observe a certain analogy with the case of algebraically closed fields. We show here in particular that connected typedefinable groups can be described as “Dalgebraic groups”, which is the natural equivalent for algebraic groups. However ..."
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When studying typedefinable sets in closed Hasse fields, we observe a certain analogy with the case of algebraically closed fields. We show here in particular that connected typedefinable groups can be described as “Dalgebraic groups”, which is the natural equivalent for algebraic groups. However, many problems arise when we try to give a description of these geometric objects in the language of schemes. We will mainly focus on the relationship between a Dring A and the Dring Â of global sections for the Dscheme defined by A. We will give some minimal conditions for these two Drings to define the same Dscheme, and exhibit an example where really bad things occur. 1
Journal of Algebra
"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit:
Model theory of fields with operators a survey
"... The model theory of fields with operators has proven to be very useful in applications of model theory to problems outside logic. Differentially closed fields and separably closed fields were instrumental in Hrushovski’s proof of the conjecture of MordellLang ([39]), and in work of Hrushovski and P ..."
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The model theory of fields with operators has proven to be very useful in applications of model theory to problems outside logic. Differentially closed fields and separably closed fields were instrumental in Hrushovski’s proof of the conjecture of MordellLang ([39]), and in work of Hrushovski and Pillay on counting the number of transcendental points on certain varieties