Research
Research Interests
Model theory and applications; more specifically, o-minimality and its interactions with other areas, including diophantine geometry, real geometry, combinatorics, topology and functional analysis.
In particular:
- Diophantine Geometry: The density of algebraic points on definable sets; Wilkie's Conjecture, both general results and special cases, and their applications, primarily to obtain transcendence results; questions of effectivity of the Pila−Wilkie Theorem and its applications.
- Real Geometry: Smooth parameterization in the definable setting of o-minimal structures, including applications of Pila−Wilkie (re)parameterization and questions of effectivity; mild parameterization in o-minimal structures and its applications.
- Definable topologies over o-minimal structures: Development of the theory of definable topological spaces; questions of metrisability, separability, regularity etc. in the definable context.
- Functional Analysis over o-minimal structures: Normed and Banach spaces over o-minimal structures, in particular function spaces; questions of universality and approximation.
- Combinatorics: Ramsey-type results in the definable setting, e.g. growth rates of Ramsey functions for definable relations.
Publications
Peer-Reviewed Publications
- Mildness and the density of rational points on certain transcendental curves
(with D. J. Miller and G. O. Jones) Notre Dame Journal of Formal Logic, 52 (1) (2011), 67-74
- An o-minimal structure without mild parameterization
Annals of Pure and Applied Logic, 162 (6) (2011), 409-418
- The density of algebraic points on certain Pfaffian surfaces
(with G. O. Jones) Quarterly Journal of Mathematics, 63 (3) (2012), 637-651
- Convergence results for functions spaces over o-minimal structures
Journal of Logic and Analysis, (4) (2012), Paper 1, 14pp.
- Integer-valued definable functions
(with G. O. Jones and A. J. Wilkie) Bulletin of the London Mathematical Society, 44 (2012), 1285–1291
- Rational values of Weierstrass zeta functions
(with G. O. Jones) Proceedings of the Edinburgh Mathematical Society (2), 59, 4 (2016), 945-958.
- Ramsey growth in some NIP structures
(with A. Chernikov and S. Starchenko) accepted to J. Inst. Math. Jussieu (2019) arXiv , 26pp.
Submitted
- Effective Pila—Wilkie bounds for surfaces implicitly defined from Pfaffian functions
(with G. O. Jones) arXiv , 32pp.
Publications In Preparation
- Directed sets and topological spaces definable in o-minimal structures
(with P. Andujar Guerrero und E. Walsberg) 28pp.
- Definable topologies
(with P. Andujar Guerrero and E. Walsberg) 65pp.
Thesis
- Convergence and Parameterization in O-Minimal Structures
D.Phil. Thesis of the University of Oxford (2009).
Funded Research Projects
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