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 …

Given an algebraic differential equation of order greater than one, it is shown that if there is
any nontrivial algebraic relation amongst any number of distinct nonalgebraic solutions …

Point-counting results for sets in real Euclidean space have found remarkable applications
to diophantine geometry, enabling significant progress on the André–Oort and Zilber–Pink …

In this paper, we prove several Ax-Schanuel type results for uniformizers of geometric
structures; our general results describe the differential algebraic relations between the …

Abstract We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the
uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on …

Motivated by the search for methods to establish strong minimality of certain low order
algebraic differential equations, a measure of how far a finite rank stationary type is from …

In this paper we develop a new technique for showing that a nonlinear algebraic differential
equation is strongly minimal based on the recently developed notion of the degree of non …

Let, for the degrees and discriminants of maximal special subvarieties; in particular, it follows
that the André–Oort conjecture for powers of the modular curve is decidable in polynomial …

We prove the Existential Closedness conjecture for the differential equation of the $ j $-
function and its derivatives. It states that in a differentially closed field certain equations …

We prove function field versions of the Zilber-Pink conjectures for varieties supporting a
variation of Hodge structures. A form of these results for Shimura varieties in the context of …