Most of my research is around differential algebra, model theory, and their applications. Over the past few years my work around differential algebra has centered around functional transcendence results. A common theme to much of this work is the application of ideas from geometric stability theory to concrete problems in algebraic differential equations, complex manifolds, transcendence and diophantine results. My student Jonathan Wolf worked in this area, and my student Matthew Devilbiss is working in this area.

Together with my student Hunter Chase, I began working on applications of model theory to machine learning. My student Kevin Zhou is working in this area.

This material is based upon work supported by the
National Science Foundation CAREER award 1945251, *Applied Model Theory*.

I have been previously been PI on NSF grants 1700095, 1834578, 1204510.

- Not Pfaffian 2021

This short note describes the connection between strong minimality of the differential equation satisfied by an complex analytic function and the real and imaginary parts of the function being Pfaffian. This connection combined with a theorem of Freitag and Scanlon (2017) provides the answer to a question of Binyamini and Novikov (2017). We also answer a question of Bianconi (2016). We give what seem to be the first examples of functions which are definable in o-minimal expansions of the reals and are differentially algebraic, but not Pfaffian.

- A differential approach to the Ax-Schanuel, I
with David Blázquez-Sanz and Guy Casale and Ronnie Nagloo. 2021

In this paper, we prove several Ax-Schanuel type results for uniformizers of geometric structures. In particular, we give a proof of the full Ax-Schanuel Theorem with derivatives for uniformizers of any Fuchsian group of the first kind and any genus. Our techniques combine tools from differential geometry, differential algebra and the model theory of differentially closed fields. The proof is very similar in spirit to Ax's proof of the theorem in the case of the exponential function.

- Algebraic relations between solutions of Painlevé equations with Joel Nagloo. 2022.

In this manuscript we make major progress classifying algebraic relations between solutions of Painlevé equations. Our main contribution is to establish the algebraic independence of solutions of various pairs of equations in the Painlevé families; for generic coefficients, we show all algebraic relations between solutions of equations in the same Painlevé family come from classically studied B{ä}cklund transformations. We also apply our analysis of ranks to establish some transcendence results for pairs of Painlevé equations from different families. In that area, we answer several open questions of Nagloo (2016), and in the process answer a question of Boalch (2012). We calculate model theoretic ranks of all Painlevé equations in this article, extending results of Nagloo and Pillay (2017). We show that the type of the generic solution of any equation in the second Painlevé family is geometrically trivial, extending a result of Nagloo (2015). We give the first model theoretic analysis of several special families of the third Painlevé equation, proving results analogous to Nagloo and Pillay (2017). We also give a novel new proof of the irreducibility of the third, fifth and sixth Painlevé equations using recent work of Freitag, Jaoui, and Moosa (2022). Our proof is fundamentally different than the existing transcendence proofs of Watanabe (1998) or Cantat and Loray (2009).

- Strong minimality of triangle functions
with Guy Casale, Matthew DeVilbiss, Joel Nagloo. 2022

In this manuscript, we give a new proof of strong minimality of certain automorphic functions, originally results of Freitag and Scanlon (2017), Casale, Freitag, and Nagloo (2020), Blázquez-Sanz, Casale, Freitag, and Nagloo (2020). Our proof is shorter and conceptually different than those presently in the literature.

- Generic differential equations are strongly minimal
with Matthew Devilbiss. Accepted at Compositio Mathematica. 2023.

In this manuscript 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 nonminimality of Freitag and Moosa. Our techniques are sufficient to show that generic order h differential equations with nonconstant coefficients are strongly minimal, answering a question of Poizat (1980).

- On the equations of Poizat and Liénard with Rémi Jaoui, David Marker, Joel Nagloo. Accepted at IMRN. 2022.

We study the structure of the solution sets in universal differential fields of certain differential equations of order two, the Poizat equations, which are particular cases of Liénard equations. We give a necessary and sufficient condition for strong minimality for equations in this class and a complete classification of the algebraic relations for solutions of strongly minimal Poizat equations. We also give an analysis of the non strongly minimal cases as well as applications concerning the Liouvillian and Pfaffian solutions of some Liénard equations. - The degree of nonminimality is at most two
with Rémi Jaoui and Rahim Moosa. Accepted at the Journal of Mathematical Logic. 2022.

It is shown that if p is a complete type of Lascar rank at least 2 over A, in the theory of differentially closed fields of characteristic zero, then there exists a pair of realisations, a1 and a2, such that p has a nonalgebraic forking extension over A,a1,a2. Moreover, if A is contained in the field of constants then p already has a nonalgebraic forking extension over A,a1. The results are also formulated in a more general setting. - When any three solutions are independent with Rémi Jaoui and Rahim Moosa. Inventiones. 2022.

Given an algebraic differential equation of order >1, it is shown that if there is any nontrivial algebraic relation amongst any number of distinct nonalgebraic solutions, along with their derivatives, then there is already such a relation between three solutions. This is deduced as an application of the following model-theoretic result: Suppose p is a stationary nonalgebraic type in the theory of differentially closed fields of characteristic zero; if any three distinct realisations of p are independent then p is minimal. An algebro-geometric formulation in terms of D-varieties is given. The same methods yield also an analogous statement about families of compact Kähler manifolds. - On the Geometry of Stable Steiner Tree Instances with Neshat Mohammadi, Aditya Potukuchi, Lev Reyzin. CCCG 2022

In this note we consider the Steiner tree problem under Bilu-Linial stability. We give strong geometric structural properties that need to be satisfied by stable instances. We then make use of, and strengthen, these geometric properties to show that 1.562-stable instances of Euclidean Steiner trees are polynomial-time solvable. We also provide a connection between certain approximation algorithms and Bilu-Linial stability for Steiner trees.

- Bounding nonminimality and a conjecture of Borovik-Cherlin
with Rahim Moosa. Journal of the European Mathematical Society. 2022.

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 being minimal is introduced and studied: The {\em degree of nonminimality} is the minimum number of realisations of the type required to witness a nonalgebraic forking extension. Conditional on the truth of a conjecture of Borovik and Cherlin on the generic multiple-transitivity of homogeneous spaces definable in the stable theory being considered, it is shown that the nonminimality degree is bounded by the U-rank plus 2. The Borovik-Cherlin conjecture itself is verified for algebraic and meromorphic group actions, and a bound of U-rank plus 1 is then deduced unconditionally for differentially closed fields and compact complex manifolds. An application is given regarding transcendence of solutions to algebraic differential equations.

- Bounds in Query Learning. with Hunter Chase. COLT 2020.

We introduce new combinatorial quantities for concept classes, and prove lower and upper bounds for learning complexity in several models of query learning in terms of various combinatorial quantities. Our approach is flexible and powerful enough to enough to give new and very short proofs of the efficient learnability of several prominent examples (e.g. regular languages and regular ω-languages), in some cases also producing new bounds on the number of queries. In the setting of equivalence plus membership queries, we give an algorithm which learns a class in polynomially many queries whenever any such algorithm exists. We also study equivalence query learning in a randomized model, producing new bounds on the expected number of queries required to learn an arbitrary concept. Many of the techniques and notions of dimension draw inspiration from or are related to notions from model theory, and these connections are explained. We also use techniques from query learning to mildly improve a result of Laskowski regarding compression schemes.

- Some functional transcendence results around the Schwarzian differential equation. with David Blázquez-Sanz and Guy Casale and Ronnie Nagloo. Annales de la Faculté des Sciences de Toulouse. 2021.

This paper centers around proving variants of the Ax-Lindemann-Weierstrass (ALW) theorem for analytic functions which satisfy Schwarzian differential equations. In previous work, the authors proved the ALW theorem for the uniformizers of genus zero Fuchsian groups, and in this work, we generalize that result in several ways using a variety of techniques from model theory, galois theory and geometry.

- Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups. with Guy Casale and Ronnie Nagloo. Annals of Mathematics. 2020.

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 differential Galois theory, monodromy of linear differential equations, the study of algebraic and Liouvillian solutions, differential algebraic work of Nishioka towards the Painlevé irreducibility of certain Schwarzian equations, and considerable machinery from the model theory of differentially closed fields.

Our techniques allow for certain generalizations of the Ax-Lindemann-Weierstrass theorem which have interesting consequences. In particular, we apply our results to answer a question of Painlevé (1895). We also answer certain cases of the André-Pink conjecture, namely in the case of orbits of commensurators of Fuchsian groups.

- Model theory and combinatorics of banned sequences
.
with Hunter Chase. Journal of Symbolic Logic.

We set up a general context in which one can prove Sauer-Shelah type lemmas. We apply our general results to answer a question of Bhaskar and give a slight improvement to a result of Malliaris and Terry. We also prove a new Sauer-Shelah type lemma in the context of op-rank, a notion of Guingona and Hill.

- Effective definability of Kolchin polynomials. with Omar Leon Sanchez and Wei Li.
Proceedings of the American Mathematical Society.

While the natural model-theoretic ranks available in differentially closed fields (of characteristic zero), namely Lascar and Morley rank, are known not to be definable in families of differential varieties; in this note we show that the differential-algebraic rank given by the Kolchin polynomial is in fact definable. As a byproduct, we are able to prove that the property of being weakly irreducible for a differential variety is also definable in families. The question of full irreducibility remains open, it is known to be equivalent to the generalized Ritt problem.

- Model theory and machine learning, with Hunter Chase. Bulletin for symbolic logic.

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.

- Finiteness theorems on
hypersurfaces in partial differential-algebraic geometry, with Rahim Moosa. Advances in Mathematics.

We establish finiteness statements for hypersurface solutions to partial differential equations with (possibly) nonconstant coefficients. We use our main theorem to answer a question of Eremenko on the height of algebraic solutions to differential equations. We also give various other applications, partially answering a question of Hrushovski and Scanlon. We also prove various bounds which are of independent interest.

- Bertini theorems for
differential
algebraic geometry Proceedings of the American Mathematical Society.

This paper is (a modified) version of a chapter of my thesis. I prove Bertini-type theorems for algebraic differential equations. The general style of the results is the following: because of various examples, we know intersections of differential algebraic varieties are complicated and display anomalous behaviour. But are these examples ubiquitous or very special? The results of this paper point towards the latter, and identify in very concrete ways how special these counterexamples are. The results proved here are also very useful for inductive proofs (inducting on dimension), and are used in my paper with Leon-Sanchez and Simmons on completeness and my paper with Li and Scanlon on differential Chow varieties.

- Differential Chow
Varieties Exist, with Wei Li and Tom
Scanlon. Journal of the London Mathematical Society.

Given an algebraic varaiety V, the collection of positive divisor of fixed degree and dimension has the structure of a constructible set - when V is projective, this constructible set is a projective variety, called the Chow variety. Chow varieties thus formalize the notion of a parameter space of subvarieties with the fixed invariants of dimension and degree. Similar numerical notions exist in the study of algebraic differential equations, and fixing these invariants, such a parameter space is called the differential Chow variety - to say that the differential Chow variety exists for a fixed set of numerical invariants came to mean that it has the stucture as a constructible set in the Kolchin topology. We show that the differential Chow variety exists, answering a question of Gao, Li, and Yuan. We also establish various bounds having to do with the degree and order of differential cycles, which are of independent interest. Will Johnson wrote an appendix in which irreducibility is shown to be definable in families in ACF via an elementary proof.

- Simple
differential field extensions and effective bounds, with Wei Li. Lecture notes in computer science.

We establish several variations on Kolchin's differential primitive element theorem, and conjecture a generalization of Pogudin's primitive element theorem. These results are then applied to improve the bounds for the effective Differential Luroth theorem. We also improve the bound for the differential Luroth theorem in several special cases, including one case with model theoretic hypotheses.

- Uniform bounding in
partial differential fields, with Omar
Leon Sanchez. Advances in Mathematics.

We prove a partial differential Bezout-type theorem, generalizing a result of Pillay and Hrushovski to the partial differential context. Various applications are discussed. In particular, applications of an effective differential nullstellensatz are described. In elementary terms,the basic problem considered is the following: bound the degree of the Zariski closure of the solution set of a system of differential equations in terms of invariants of the system (order, degree, and number of equations).

- Superstability and
Central Extensions of Algebraic Groups, with Andrey Minchenko. Advances in Applied
Mathematics.

Altinel and Cherlin proved that any perfect central extension of a simple algebraic group over an algebraically closed field which happens to be of finite Morley rank is actually a finite central extension and is itself an algebraic group. We will prove an infinite rank version of their result with an additional hypothesis, while giving an example which shows the necessity of this hypothesis. The inspiration for the work comes from differential algebra; namely, a differential algebraic version of the results here was used by the second author to answer a question of Cassidy and Singer. The work here also provides an alternate path to the same answer.

- Strong minimality and
the
j-function, with Tom
Scanlon. Journal of the EMS.

The classical analytic j-function satisfie a third order (nonlinear) differential equation. In this paper, we show how a model theoretic analysis of this differential equation can be used to settle an old open problem in differential algebra and prove special points conjectures around the Andre-Oort conjecture. There are three main contributiuons of the paper. We completely describe the geometry or model theory of the solution set of the differential equation satisfied by the j-function, along with the analytic covering maps of other arithmetic quotients of the upper half plane. We use this description to answer an old open problem in differential algebra (first proposed by Lascar and Poizat). We then use our model theoretic results to answer a question of Mazur about elliptic curves, and prove special points conjectures around the Andre-Oort conjecture in an effective manner.

In case you looked at the first version of this paper, it has been significantly expanded to include an analysis of the fibers of the differential rational function of order three which gives the locus of the j-function over the complex numbers. The original paper included only an analysis of the fibre above zero. This analysis has various arithmetic consequences; in particular, giving uniform bounds for the intersection of non-weakly special subvarieties with isogeny classes in products of modular curves. In addition to the tools used in the original argument (analytic embedding of differential fields, the Ax-Lindemann-Weierstrass with derivatives theorem, Nishioka's theory of automorphic functions, and indiscernible sequences), the new analysis uses ananlytic geometry and a Baire category style argument.

- On completeness and
independence for differential algebraic
varieties, with Omar
Leon Sanchez and William
Simmons. Communications in Algebra.

We extend Kolchin's results on linear dependence over projective varieties in the constants, to linear dependence over arbitrary complete differential varieties. We show that in this more general setting, the notion of linear dependence still has necessary and sufficient conditions given by the vanishing of a certain system of differential polynomials equations. We also discuss some conjectural questions around completeness and the catenary problem.

- Indecomposability for
differential algebraic groups, Journal
of
Pure and Applied Algebra.

We study a notion of indecomposability in differential algebraic groups which is inspired by both model theory and differential algebra. After establishing some basic definitions and results, we prove an indecomposability theorem for differential algebraic groups. The theorem establishes a sufficient criterion for the subgroup of a differential algebraic group generated by an infinite family of subvarieties to be a differential algebraic subgroup. This theorem is used for various definability results. For instance, we show that every noncommutative almost simple differential algebraic group is perfect, solving a problem of Cassidy and Singer. We also establish numerous bounds on Kolchin polynomials, some of which seem to be of a nature not previously considered in differential algebraic geometry; in particular, we establish bounds on the Kolchin polynomial of the generators of the differential field of definition of a differential algebraic variety.

- Completeness for partial
differential fields, Journal
of Algebra.

This paper is part of the model theory of fields of characteristic 0, equipped with m commuting derivation operators. It continues to partial differential fields work begun by Wai-Yan Pong, who treated the case m=1. We study the concept of completeness in differential algebraic geometry, applying methods of model theory and differential algebra. Our central tool in applying the valuative criterion developed in differential algebra by E.R. Kolchin, Peter Bloom, and Sally Morrison is a theorem due to Lou van den Dries. We use this valuative criterion to give a new family of complete differential algebraic varieties. In addition to completeness, we prove some embedding theorems for differential algebraic varieties of arbitrary differential transcendence degree. As a special case, we show that every differential algebraic subvariety of the projective line which has Lascar rank less than $omega^m$ can be embedded in the affine line.

- Isogeny
in superstable groups, Archive for Mathematical Logic.

We study and develop a notion of isogeny for superstable groups inspired by the notion in algebraic groups and differential algebraic notions developed by Cassidy and Singer. We prove several fundamental properties of the notion. Then we use it to formulate and prove a uniqueness results for a decomposition theorem about superstable groups similar to one proved by Baudisch. Connections to existing model theoretic notions and existing differential algebraic notions are explained.