1,719 research outputs found
Groupoids, imaginaries and internal covers
Let be a first-order theory. A correspondence is established between
internal covers of models of and definable groupoids within . We also
consider amalgamations of independent diagrams of algebraically closed
substructures, and find strong relation between: covers, uniqueness for
3-amalgamation, existence of 4-amalgamation, imaginaries of T^\si, and
definable groupoids. As a corollary, we describe the imaginary elements of
families of finite-dimensional vector spaces over pseudo-finite fields.Comment: Local improvements; thanks to referee of Turkish Mathematical
Journal. First appeared in the proceedings of the Paris VII seminar:
structures alg\'ebriques ordonn\'ee (2004/5
Definable equivalence relations and zeta functions of groups
We prove that the theory of the -adics admits elimination
of imaginaries provided we add a sort for for each . We also prove that the elimination of
imaginaries is uniform in . Using -adic and motivic integration, we
deduce the uniform rationality of certain formal zeta functions arising from
definable equivalence relations. This also yields analogous results for
definable equivalence relations over local fields of positive characteristic.
The appendix contains an alternative proof, using cell decomposition, of the
rationality (for fixed ) of these formal zeta functions that extends to the
subanalytic context.
As an application, we prove rationality and uniformity results for zeta
functions obtained by counting twist isomorphism classes of irreducible
representations of finitely generated nilpotent groups; these are analogous to
similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald
for subgroup zeta functions of finitely generated nilpotent groups.Comment: 89 pages. Various corrections and changes. To appear in J. Eur. Math.
So
On finite imaginaries
We study finite imaginaries in certain valued fields, and prove a conjecture
of Cluckers and Denef.Comment: 15p
Ample Pairs
We show that the ample degree of a stable theory with trivial forking is
preserved when we consider the corresponding theory of belles paires, if it
exists. This result also applies to the theory of -structures of a trivial
theory of rank .Comment: Research partially supported by the program MTM2014-59178-P. The
second author conducted research with support of the programme
ANR-13-BS01-0006 Valcomo. The third author would like to thank the European
Research Council grant 33882
Ample Pairs
We show that the ample degree of a stable theory with trivial forking is
preserved when we consider the corresponding theory of belles paires, if it
exists. This result also applies to the theory of -structures of a trivial
theory of rank .Comment: Research partially supported by the program MTM2014-59178-P. The
second author conducted research with support of the programme
ANR-13-BS01-0006 Valcomo. The third author would like to thank the European
Research Council grant 33882
Non-archimedean tame topology and stably dominated types
Let be a quasi-projective algebraic variety over a non-archimedean valued
field. We introduce topological methods into the model theory of valued fields,
define an analogue of the Berkovich analytification of ,
and deduce several new results on Berkovich spaces from it. In particular we
show that retracts to a finite simplicial complex and is locally
contractible, without any smoothness assumption on . When varies in an
algebraic family, we show that the homotopy type of takes only a
finite number of values. The space is obtained by defining a
topology on the pro-definable set of stably dominated types on . The key
result is the construction of a pro-definable strong retraction of
to an o-minimal subspace, the skeleton, definably homeomorphic to a space
definable over the value group with its piecewise linear structure.Comment: Final versio
Model Theory: groups, geometry, and combinatorics
This conference was about recent interactions of model theory with combinatorics, geometric group theory and the theory of valued fields, and the underlying pure model-theoretic developments. Its aim was to report on recent results in the area, and to foster communication between the different communities
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