Parameterization and Algebraic Points in O-Minimal Structures

• Applicant: Professorin Dr. Margaret E.M. Thomas
• Subject Area: Mathematics
• Term: from 2012 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 234252392

This research project is directed towards a conjecture of Wilkie lying at the intersection of model theory and diophantine geometry. Wilkie's Conjecture suggests a significant improvement, for the real exponential field, to the seminal theorem of Pila and Wilkie bounding the density of algebraic points on sets definable in o-minimal structures. This program of research will focus on the main geometric technique already being exploited in approaches to Wilkie's Conjecture, namely the mild parameterization of sets definable in o-minimal structures. The bound of Pila and Wilkie is the best bound already known for the density of algebraic points on sets definable in o-minimal structures in general. The work of Masser, Pila, Tsimerman and Zannier has shown that this bound has important consequences for major number theoretic questions, providing a new proof of the Manin--Mumford Conjecture and the first proofs not dependent on the Generalised Riemann Hypothesis of particular cases of the André--Oort Conjecture. Wilkie's Conjecture would improve the subpolynomial Pila--Wilkie bound to a logarithmic one for sets definable in the (o-minimal) real exponential field.We will address questions about mild parameterization in several related o-minimal expansions of the real field. The intention is to advance the theory of these o-minimal structures as well as to provide insights into Wilkie's Conjecture.One aim is to make progress towards establishing mild parameterization for the real exponential field. This will directly address several questions about Wilkie's Conjecture, e.g. mild parameterization for surfaces would immediately prove the conjecture for surfaces. Mild parameterization for specific examples of definable sets can also establish transcendence results.We will also consider mild parameterization in three further settings: the real field expanded by (a) restricted analytic functions; (b) certain Gevrey functions (these are mild functions, and play a role in dynamical systems); and (c) certain classes of functions whose germs at zero form a quasianalytic class (this generalises the previous case, and has connections to the theory of differential equations). In the first setting, the goal is to analyse in detail a result of Jones, Miller and Thomas, which shows that all definable sets here have mild parameterization; we aim to gain control over the constants appearing in the bounds on the parameterizing functions. In the second and third cases, the goal is to establish, using analogous methods - analysing model completeness constructions - whether or not the definable sets in these structures have mild parameterization.

• DFG Programme: Research Grants
• International Connection: Portugal, United Kingdom
• Participating Persons: Dr. Gareth Jones; Dr. Tamara Servi; Professor Dr. Alex J. Wilkie