Sharply o-minimal Structures: towards a theory of arithmetically tame geometry

ongoing

"O-minimality is a model-theoretic formalism of tame geometry. Sets that are definable in o-minimal structures enjoy strong finiteness properties, such as the existence of finite stratifications and triangulations. While drawing inspiration from the classical areas of semialgebraic and subanalytic geometry, o-minimality encompasses a strictly larger range of structures - most notably structures defined using the logarithmic and exponential functions. In the past 15 years o-minimality has enjoyed a golden age, as deep connections relating these larger structures to arithmetic geometry and Hodge theory have been unfolding. However, over this period it has become clear that some finer aspects of tameness, especially as it relates to arithmetic, are not accessible in the full generality of o-minimal theory. Some prominent conjectures have been formulated only for specific structures, with a folklore expectation that they should hold in all structures naturally arising in algebraic and arithmetic geometry. In this project we propose to refine the foundation of o-minimal geometry by introducing a notion of ""sharply o-minimal structures'', with the goal of capturing the finer arithmetic properties of the definable sets arising in algebraic and arithmetic geometry. We argue that this should be achieved by postulating sharper estimates for the asymptotic interaction between definable and algebraic sets. The construction of such ""sharp"" structures has until recently seemed technically unattainable, but three recent technical developments, including the first example of a sharply o-minimal structure beyond the semialgebraic case, renders the project timely and potentially feasible. We show how many recent advances in the area point to sharp o-minimality as a possible grand unifying framework, and illustrate how a realization of this program would greatly simplify, strengthen and generalize many of the state of the art applications of o-minimality."