EffectiveTG · Effective Methods in Tame Geometry and Applications in Arithmetic and Dynamics

Tame geometry studies structures in which every definable set has afinite geometric complexity. The study of tame geometry spans severalinterrelated mathematical fields, including semialgebraic,subanalytic, and o-minimal geometry. The past decade has seen theemergence of a spectacular link between tame geometry and arithmeticfollowing the discovery of the fundamental Pila-Wilkie countingtheorem and its applications in unlikely diophantineintersections. The P-W theorem itself relies crucially on theYomdin-Gromov theorem, a classical result of tame geometry withfundamental applications in smooth dynamics.It is natural to ask whether the complexity of a tame set can beestimated effectively in terms of the defining formulas. While a largebody of work is devoted to answering such questions in thesemialgebraic case, surprisingly little is known concerning moregeneral tame structures - specifically those needed in recentapplications to arithmetic. The nature of the link between tamegeometry and arithmetic is such that any progress toward effectivizingthe theory of tame structures will likely lead to effective resultsin the domain of unlikely intersections. Similarly, a more effectiveversion of the Yomdin-Gromov theorem is known to imply importantconsequences in smooth dynamics.The proposed research will approach effectivity in tame geometry froma fundamentally new direction, bringing to bear methods from thetheory of differential equations which have until recently never beenused in this context. Toward this end, our key goals will be to gaininsight into the differential algebraic and complex analytic structureof tame sets; and to apply this insight in combination with resultsfrom the theory of differential equations to effectivize key resultsin tame geometry and its applications to arithmetic and dynamics. Ibelieve that my preliminary work in this direction amply demonstratesthe feasibility and potential of this approach.