This book chapter is published open access.
The theory of o-minimal structures provides a powerful framework for the study of geometrically tame structures. In the past couple of decades a deep link connecting o-minimality to algebraic and arithmetic geometry has been developing. It has been clear, however, that the axioms of o-minimality do not fully capture some algebro-arithmetic aspects of tameness that one may expect in structures arising from geometry. We propose a notion of sharply o-minimal structures refining the standard axioms of o-minimality, and outline through conjectures and various partial results the potential development of this theory in parallel to the standard one.
We illustrate some applications of this emerging theory in two main directions. First, we show how it can be used to deduce Galois orbit lower bounds—notably including in nonabelian contexts where the standard transcendence methods do not apply. Second, we show how it can be used to derive effectivity and (polynomial-time) computability results for various problems of unlikely intersection around the Manin–Mumford, André–Oort, and Zilber–Pink conjectures.