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.

Abstract. The Chow variety is a parameter space for effective algebraic cycles on P n (or A n ) of given dimension and degree. We construct its analog for differential algebraic cycles on A n , answering a question of [12]. The proof uses the construction of classical algebro-geometric Chow varieties, the theory of characteristic sets of differential varieties and algebraic varieties, the theory of prolongation spaces, and the theory of differential Chow forms. In the course of the proof several definability results from the theory of algebraically closed fields are required. Elementary proofs of these results are given in the appendix.

We study completeness in partial differential varieties. We generalize many of the results of Pong to the partial differential setting. In particular, we establish a valuative criterion for differential completeness and use it to give a new class of examples of complete partial differential varieties. In addition to completeness we prove some new embedding theorems for differential algebraic varieties. We use methods from both differential algebra and model theory.Completeness is a fundamental notion in algebraic geometry. In this paper, we examine an analogue in differential algebraic geometry. The paper builds on Pong's [16] and Kolchin's [7] work on differential completeness in the case of differential varieties over ordinary differential fields and generalizes to the case differential varieties over partial differential fields with finitely many commuting derivations. Many of the proofs generalize in the straightforward manner, given that one sets up the correct definitions and attempts to prove the correct analogues of Pong's or Kolchin's results. Of course, some of the results are harder to prove because our varieties may be infinite differential transcendence degree. Nevertheless, Proposition 3.3 generalizes a theorem of [17] even when we restrict to the ordinary case. The proposition also generalizes a known result projective varieties. Many of the results are model-theoretic in nature or use model-theoretic tools.The model theory of partial differential fields was developed in [10]. For a recent alternate (geometric) axiomatization of partial differentially closed fields, see [18]. For a reference in differential algebra, we suggest [8] and [9]. The differential varieties we consider will be embedded in projective space. Generalizations to differential schemes are of interest, but are not treated here. Pillay also considers differential completeness for a slightly different category in [14]. Though Pillay's conditions for differential completeness are implied by the conditions here, their precise relationship is not clear.

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