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

Description du projet

Classe spéciale de structures o-minimales à l’étude

La o-minimalité est un formalisme théorique de la géométrie modérée. Les ensembles définissables dans des structures o-minimales possèdent de fortes propriétés de finitude, telles que l’existence de stratifications et de triangulations finies. Cependant, certains aspects plus fins de la modération, notamment en ce qui concerne l’arithmétique, ne sont pas accessibles dans toute la généralité de la théorie o-minimale. Le projet SharpOS, financé par le CER, introduit la notion de «structures fortement o-minimales» pour capturer les propriétés arithmétiques les plus fines des ensembles définissables issus de la géométrie algébrique et arithmétique. Les travaux du projet seront basés sur des avancées récentes dans la construction de telles structures pointues, comme le premier exemple de structure fortement o-minimale au-delà du cas semi-algébrique.

Objectif

"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."