Multidimensional exact classes and Lie coordinatisation
Model theory of finite and pseudofinite structures, University of Leeds, 29th July 2016
Abstract:
We introduce the notion of a multidimensional exact class, as jointly developed with Anscombe, Macpherson and Steinhorn. We then sketch a proof of the following result: For any countable language $\mathcal{L}$ and for any positive $d\in\mathbb{N}$, the class $\mathcal{C}(\mathcal{L},d)$ of all finite $\mathcal{L}$-structures with at most $d$ 4-types forms a multidimensional exact class. The proof makes extensive use of smooth approximation, a notion introduced by Lachlan in the 1980s, and the equivalent notion of Lie coordinatisation, as deeply developed by Cherlin and Hrushovski in their book Finite Structures with Few Types (Princeton, 2003).
We introduce the notion of a multidimensional exact class, as jointly developed with Anscombe, Macpherson and Steinhorn. We then sketch a proof of the following result: For any countable language $\mathcal{L}$ and for any positive $d\in\mathbb{N}$, the class $\mathcal{C}(\mathcal{L},d)$ of all finite $\mathcal{L}$-structures with at most $d$ 4-types forms a multidimensional exact class. The proof makes extensive use of smooth approximation, a notion introduced by Lachlan in the 1980s, and the equivalent notion of Lie coordinatisation, as deeply developed by Cherlin and Hrushovski in their book Finite Structures with Few Types (Princeton, 2003).