Model theory of multidimensional asymptotic classes

• Main Author: Wolf, Daniel Anthony
• Other Authors: Macpherson, H. Dugald
• Published: University of Leeds 2016
• Subjects: 515
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.695968

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.