Abstract. We show that the order three algebraic differential equation over Q satisfied by the analytic j-function defines a non-ℵ 0 -categorical strongly minimal set with trivial forking geometry relative to the theory of differentially closed fields of characteristic zero answering a long-standing open problem about the existence of such sets. The theorem follows from Pila's modular Ax-Lindemann-Weierstrass with derivatives theorem using Seidenberg's embedding theorem and a theorem of Nishioka on the differential equations satisfied by automorphic functions. As a by product of this analysis, we obtain a more general version of the modular Ax-Lindemann-Weierstrass theorem, which, in particular, applies to automorphic functions for arbitrary arithmetic subgroups of SL 2 (Z). We then apply the results to prove effective finiteness results for intersections of subvarieties of products of modular curves with isogeny classes. For example, we show that if ψ : P 1 → P 1 is any non-identity automorphism of the projective line and t ∈ A 1 (C) A 1 (Q alg ), then the set of s ∈ A 1 (C) for which the elliptic curve with j-invariant s is isogenous to the elliptic curve with j-invariant t and the elliptic curve with j-invariant ψ(s) is isogenous to the elliptic curve with j-invariant ψ(t) has size at most 367 . In general, we prove that if V is a Kolchin-closed subset of A n , then the Zariski closure of the intersection of V with the isogeny class of a tuple of transcendental elements is a finite union of weakly special subvarieties. We bound the sum of the degrees of the irreducible components of this union by a function of the degree and order of V .

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Motivated by the effective bounds found in [12] for ordinary differential equations, we prove an effective version of uniform bounding for fields with several commuting derivations. More precisely, we provide an upper bound for the size of finite solution sets of partial differential polynomial equations in terms of data explicitly given in the equations and independent of parameters. Our methods also produce an upper bound for the degree of the Zariski closure of solution sets, whether they are finite or not.

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About 25 years ago, it came to light that a single combinatorial property determines both an important dividing line in model theory (NIP) and machine learning (PAC-learnability). The following years saw a fruitful exchange of ideas between PAC-learning and the model theory of NIP structures. In this article, we point out a new and similar connection between model theory and machine learning, this time developing a correspondence between stability and learnability in various settings of online learning. In particular, this gives many new examples of mathematically interesting classes which are learnable in the online setting.