- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

- 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).

- 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.

- 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.

- 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.

- 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.

- 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.

- 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. However, Ruizhang Jin
and I are working to fix an error that Ruizhang recently discovered. We
expect a revision shortly in which Ruizhang Jin will also be a
co-author.

- 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.

- 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.

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

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

- Effective definability of Kolchin polynomials. with Omar Leon Sanchez and Wei Li. 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.

Preprints

- 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 $F.$

- Co-analysis
and inertial dimension, with Will Johnson, Alice Medvedev, and Tom Scanlon. Notes for the geometry
of the Frobenius meeting.

- 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.

- 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.