EffectiveTG SIGNED

Project "EffectiveTG" data sheet

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 Project objective

Tame geometry studies structures in which every definable set has a finite geometric complexity. The study of tame geometry spans several interrelated mathematical fields, including semialgebraic, subanalytic, and o-minimal geometry. The past decade has seen the emergence of a spectacular link between tame geometry and arithmetic following the discovery of the fundamental Pila-Wilkie counting theorem and its applications in unlikely diophantine intersections. The P-W theorem itself relies crucially on the Yomdin-Gromov theorem, a classical result of tame geometry with fundamental applications in smooth dynamics.

It is natural to ask whether the complexity of a tame set can be estimated effectively in terms of the defining formulas. While a large body of work is devoted to answering such questions in the semialgebraic case, surprisingly little is known concerning more general tame structures - specifically those needed in recent applications to arithmetic. The nature of the link between tame geometry and arithmetic is such that any progress toward effectivizing the theory of tame structures will likely lead to effective results in the domain of unlikely intersections. Similarly, a more effective version of the Yomdin-Gromov theorem is known to imply important consequences in smooth dynamics.

The proposed research will approach effectivity in tame geometry from a fundamentally new direction, bringing to bear methods from the theory of differential equations which have until recently never been used in this context. Toward this end, our key goals will be to gain insight into the differential algebraic and complex analytic structure of tame sets; and to apply this insight in combination with results from the theory of differential equations to effectivize key results in tame geometry and its applications to arithmetic and dynamics. I believe that my preliminary work in this direction amply demonstrates the feasibility and potential of this approach.