Model Theory, Geometric Structures and Special Functions

In this project, the investigator seeks to significantly expand his work on applications of model theory, a branch of mathematical logic, to tackle several far-reaching open problems centered around the study of geometric structures. Roughly speaking, a geometric structure modeled on a geometry (G,X) is a space that locally (i.e. upon zooming in) looks like the given space X and on which there are local symmetries coming from the action of the group G on X. These structures have been studied extensively since their introduction by Ehresmann in the 1930’s and in this project the PI aims to continue to explore their connections to number theory and differential equations. The project will involve training of graduate students. The main objective is to make progress on fundamental questions about the algebraic nature of the wide class of special functions coming from geometric structures (as well as from other physical applications) and to use those to attack related problems in diophantine geometry. More precisely, building on his recent work on a differential approach to functional transcendence, the investigator aims to 1) tackle major transcendence problems such as the Ax-Lindemann-Weierstrass and Ax-Schanuel conjectures for uniformizing functions of geometric structures in higher dimensions, 2) initiate the study of the Existential Closedness and the Zilber-Pink conjecture for Fuchsian functions using a differential approach and 3) study the structure of the sets defined by (possibly other) classical differential and difference equations. 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.