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Model theory of multidimensional asymptotic classes

Wolf, Daniel Anthony (2016) Model theory of multidimensional asymptotic classes. PhD thesis, University of Leeds.

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Abstract

In this PhD thesis we explore the concept of a multidimensional asymptotic class. This is a new notion in model theory, arising as a generalisation of the Elwes–Macpherson–Steinhorn notion of an N-dimensional asymptotic class [22] and thus ultimately as a development of the Lang–Weil estimates of the number of points of a variety in a finite field [47]. We provide the history and motivation behind the topic before developing its basic theory, paying particular attention to multidimensional exact classes, a special kind of multidimensional asymptotic class where the measuring functions provide the precise sizes of the definable sets, rather than only approximations. We describe a number of examples and non-examples and then show that multidimensional asymptotic classes are closed under bi-interpretability. We use results about smoothly approximable structures [35] and Lie coordinatisable structures [18] to prove the following result, as conjectured by Macpherson: For any countable language L and any positive integer d the class C(L,d) of all finite L-structures with at most d 4-types is a polynomial exact class in L; here a polynomial exact class is a multidimensional exact class with polynomial measuring functions. We finish the thesis by posing some open questions, indicating potential further lines of research.

Item Type: Thesis (PhD)
Keywords: model theory, asymptotic classes, smooth approximation, Lie coordinatisation
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.695968
Depositing User: Daniel Wolf
Date Deposited: 31 Oct 2016 13:18
Last Modified: 25 Jul 2018 09:53
URI: http://etheses.whiterose.ac.uk/id/eprint/15316

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