# James Freitag - Research

## My research:

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. Together with my student Hunter Chase, I have been working on applications of model theory to machine learning.

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.

## Thesis

Model Theory and Differential Algebraic Geometry, PhD thesis, University of Illinois at Chicago, August 2012.

## Published papers

1. Isogeny in superstable groups, Archive for Mathematical Logic, 2014.
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.

2. Completeness for partial differential fields, Journal of Algebra, 2014.
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.

3. Indecomposability for differential algebraic groups, Journal of Pure and Applied Algebra, 2015.
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.

4. On completeness and independence for differential algebraic varieties, with Omar Leon Sanchez and William Simmons. Communications in Algebra, 2015.
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.

5. Strong minimality and the j-function, with Tom Scanlon. Journal of the EMS, 2015.
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.

6. Superstability and Central Extensions of Algebraic Groups, with Andrey Minchenko. Advances in Applied Mathematics, 2015.
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.

7. 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.

8. 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).

9. 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.

10. 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.

11. 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.

12. Model theory and machine learning, with Hunter Chase.  To appear: 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.

13. 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.

14. 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.

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

16. 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.
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.

17. 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.

## Preprints:

1. VC-minimality: examples and observations, with Uri Andrews, Sarah Cotter Blanset, and Alice Medvedev We prove that Presberger arithmetic is not VC-minimal, and answer various questions of Adler and Guingona.

2. Disintegrated order one differential equations and algebraic general solutions, with an appendix jointly with Ronnie Nagloo and Ngoc Thieu Vo. We generalize results of Rosenlicht to give a necessary and sufficient condition for when order one differential equations of the form D(x)=f(x) where f is a rational function is orthogonal to the constants. Following the main results of the paper, we also explain the connection between algebraic general solutions and weak orthogonality to the constants. The results of this paper are used (in a separate work) to answer a question of Hrushovski and Scanlon on Lascar rank and Morley rank in differential fields. This paper has been recommended for publication pending some minor revisions, in the Israel Journal of Mathematics.
3. Order one differential equations on nonisotrivial algebraic curves, with Taylor Dupuy and Aaron Royer. In this paper we provide new examples of geometrically trivial strongly minimal differential algebraic varieties living on nonisotrivial curves over differentially closed fields of characteristic zero. These are systems whose solutions only have binary algebraic relations between them. Our technique involves developing a theory of τ-forms, and building connections to deformation theory. This builds on previous work of Buium and Rosen. In our development, we answer several open questions posed by Rosen and Hrushovski-Itai.

4. Painleve equations, vector fields, and ranks in differential fields. Model theoretic ranks of solutions to Painleve equations are calculated, and the type of the generic solution of the second Painleve equation is shown to be disintegrated, strengthening a theorem of Nagloo. A question of Hrushovski and Scanlon regarding Lascar rank and Morley rank in differential fields is answered using planar vector fields.

5. Groups of small typical differential dimension. We apply techniques from $\omega$-stable and superstable groups to strongly connected and almost simple differential algebraic groups in the sense of Cassidy and Singer. We analyze differential algebraic group actions from this point of view, and prove several results regarding interpreting fields from these actions. We prove a differential algebraic analogue of Rienecke's theorem. We show that every strongly connected differential algebraic group with typical differential dimension two is solvable. A special instance of the Cassidy-Singer conjecture is confirmed. Namely, noncommutative almost simple groups of typical differential dimension $3$ are equal to $SL_2(F)$ or $PSL_2(F)$ for a definable subfield $F.$

## Informal notes

1. Co-analysis and inertial dimension, with Will Johnson, Alice Medvedev, and Tom Scanlon. Notes for the geometry of the Frobenius meeting.
2. Generics in differential fields. When I was in my first year of graduate school, Dave Marker asked me for an example in the Kolchin topology such that an open subset had lower Morley rank than its closure. This also answered a related question posed by Benoist.
3. Relative Kolchin irreducibility. This is a note about how to prove the affine Kolchin irreducibility theorem for the partial differential case - this means an irreducible differential algebraic variety remains irreducible when adding derivatives to the distinuished set of derivations. The case that is often cited occurs when one starts with an algebraic variety defined over the constants. In this case, there are multiple proofs using geometric techniques (resolution of singularities and weaker forms of resolution). This note explains how to derive the full theorem from that special case.