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 Ndimensional 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 nonexamples and then show that multidimensional asymptotic classes are closed under biinterpretability. 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 Lstructures with at most d 4types 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 