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Non-archimedean stratifications in T-convex fields.

Garcia Ramirez, Erick (2017) Non-archimedean stratifications in T-convex fields. PhD thesis, University of Leeds.

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Abstract

We prove that whenever T is a power-bounded o-minimal theory, t-stratifications exist for definable maps and sets in T-convex fields. To this effect, a thorough analysis of definability in T-convex fields is carried out. One of the conditions required for the result above is the Jacobian property, whose proof in this work is a long and technical argument based on an earlier proof of this property for valued fields with analytic structure. An example is given to illustrate that t-stratifications do not exist in general when T is not power-bounded. We also show that if T is power-bounded, the theory of all T-convex fields is b-minimal with centres. We also address several applications of tstratifications. For this we exclusively work with a power-bounded T. The first application establishes that a t-stratification of a definable set X in a T-convex field induces t stratifications on the tangent cones of X. This is a contribution to local geometry and singularity theory. Regarding R as a model of T, the remaining applications are derived by considering the stratifications induced on R by t-stratifications in non-standard models. We prove that each such induced stratification is a C1-Whitney stratification; this in turn leads to a new proof of the existence of Whitney stratifications for definable sets in R. We also deal with interactions between tangent cones of definable sets in R and stratifications.

Item Type: Thesis (PhD)
Keywords: Model theory, valued fields, t-stratifications, o-minimality, T-convex fields, weak ominimality, b-minimality, tangent cones, Jacobian property, archimedean t-stratifications, Whitney stratifications.
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds)
Identification Number/EthosID: uk.bl.ethos.725009
Depositing User: Erick Garcia Ramirez
Date Deposited: 23 Oct 2017 15:21
Last Modified: 25 Jul 2018 09:56
URI: http://etheses.whiterose.ac.uk/id/eprint/18362

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