Parameterization and Algebraic Points in O-Minimal Structures
Final Report Abstract
This project resulted in many new insights concerning the application of o-minimality to diophantine geometry. It was directed towards a bold conjecture lying at this intersection, namely Wilkie’s Conjecture, which suggests a significant improvement (for the real exponential field) to the seminal counting theorem of Pila and Wilkie. The central goal of the research project was to improve our understanding of the main geometric technique being exploited in approaches to Wilkie’s Conjecture, namely the mild parameterization of sets definable in o-minimal structures, as well as to consider related questions of o-minimal parameterization and its applications to diophantine geometry, and the nature of sets definable in the real exponential field. Many new results concerning mild parameterization in the o-minimal setting were proved, in particular as part of a successful doctoral research project that was funded through this project. Focussing on o-minimal expansions of the real field by quasianalytic classes resulted in several such structures being shown to have definable mild parameterization. These include the real field expanded by Gevrey functions (which play a role in dynamical systems), which moreover resulted in a diophantine application. Contributions were also made towards understanding the connection between mild functions and ultradifferentiable functions that lie in quasianalytic Denjoy–Carleman (QADC) classes. Such QADC classes containing mild functions were shown to be closed under differentiation, which led to further new examples of o-minimal structures with definable mild parameterization. By contrast, a new class of examples of o-minimal structures was found which do not have definable mild parameterization, namely polynomially bounded o-minimal structures that define any irrational power function. Nevertheless, one-dimensional sets definable in the expansion of the real field by power functions were shown to have (non-definable) mild parameterization. Through this compendium of results, this project has significantly advanced our understanding of mild parameterization in the o-minimal setting. Moreover, the study of sets definable in the real exponential field that was made through this project led to a greater understanding of their combinatorial nature, and in particular resulted in a surprising characterization of o-minimal structures in terms of bounds on Ramsey numbers. In addition, a complementary study of the role of o-minimal k-parameterization in the Pila–Wilkie Theorem led to a uniform effective k-parameterization result for surfaces implicitly defined from restricted Pfaffian functions, and hence an effective version of the Pila–Wilkie Theorem for surfaces implicitly defined from (total) Pfaffian functions. This promises to have further applications to effective results in diophantine geometry. This project was furthermore supported by the hosting of the Workshop: O-Minimality and Applications, held at the University of Kontanz in July 2015. This provided great impulse to the project as well as to the work of researchers active in related areas.
Publications
- (2021) Ramsey growth in some NIP structures. Journal of the Institute of Mathematics of Jussieu, 20 (1) 1-29
Chernikov, Artem; Starchenko, Sergei; Thomas, Margaret E. M.
(See online at https://doi.org/10.1017/S1474748019000100) - Effective Pila–Wilkie bounds for unrestricted Pfaffian surfaces. 2018
Gareth O. Jones and Margaret E. M. Thomas
- Mild parameterization in o-minimal structures. Dissertation, Naturwissenschaften (Dr.rer.nat), Universität Konstanz. 2019
Derya Çıray
