Model Theory: Connecting Algebraic, Analytic, and Diophantine Geometry Through Definability

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Important mathematical structures can often be understood from very different perspectives, either by analyzing them with algebraic or analytic formulas or by regarding them geometrically. This dual algebraic/geometric view is applied in many branches of mathematics, especially for the study of algebraic equations involving numbers (under the name of Diophantine geometry), the study of solutions to polynomial equations (under the name of algebraic geometry), or the study of the possible algebraic relations among solutions to systems of differential or difference equations (under the names of differential algebraic geometry or difference algebraic geometry, respectively), among others. In practice, some of the questions considered in these areas suffer from extreme complexity inherited from their connection to number theory or to even more complicated domains. Work in mathematical logic has elucidated the boundary between those complicated theories and those admitting a tame, geometric theory. This research project aims to extend the class of theories for which a tame geometry can be established and to use these results from mathematical logic to answer questions from the target domains. This research project studies the connections between geometries of various kinds, including algebraic, differential, Diophantine, and analytic geometries, through the model-theoretic lens of definability in suitable theories. Mathematical theories as diverse as those of partial differential equations, difference equations, perfectoid spaces, formal geometry, algebraic dynamics, and homogenous dynamics will be studied. Technically, methods including geometric stability theory as applied to differentially closed fields, o-minimality, and quantifier elimination for valued differential fields and analytic difference rings will be applied for the purpose of answering questions internal to model theory (such as proving or refuting the trichotomy principle for regular types in differentially closed fields with several commuting derivations) and for applications to problems in functional transcendence, dynamics, and Diophantine geometry. It is anticipated that the work will have applications to the structure of algebraic differential equations, the arithmetic of dynamical systems, and mathematical physics, as well as in other areas.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.