# Blog Archives

These are the items posted in this seminar, currently ordered by their post-date, rather than by the event date. We will create improved views in the future. In the meantime, please click on the Seminar menu item above to find the page associated with this seminar, which does have a more useful view order.# Imaginaries in valued differential fields I: Finding prolongations

In 2000, Scanlon described a theory of existentially closed differential fields where the derivation is contractive: v(d(x)) ≥ v(x), for all x. He also proved a quantifier elimination result for this theory. Around the same time, Haskell, Hrushovski and Macpherson classified all the quotients of definable sets by definable equivalence relations in a algebraically closed valued field by proving elimination of imaginaries (relative to certain quotients of the linear group). In analogy with the pure field situation where elimination of imaginaries for differentially closed fields can be derived from elimination of imaginaries in the underlying algebraically closed field, it was conjectured that Scanlon’s theory of existentially closed contractive valued differential fields also eliminated imaginaries relatively to those same quotients of the linear group.

In this talk, I will describe the first part of the proof that this result indeed holds. Our main goal will be to explain a construction that can be interpreted as finding “generic” prolongations for valued differential constructible sets.

# An introduction to Pillay’s differential Galois theory (part 2)

In a series of papers from the 1990s and early 2000s, Pillay used the machinery of model-theoretic binding groups to give a slick geometric account and generalization of Kolchin’s theory of strongly normal extensions and constrained cohomology. This series of two talks is intended to be expository, with its main goal being to introduce and frame the relevant model-theoretic notions of internality and binding groups within the context of differential algebra, as well as to go through Pillay’s argument that his generalized strongly normal extensions arise from logarithmic differential equations defined over algebraic *D*-groups.

# An introduction to Pillay’s differential Galois theory (part 1)

In a series of papers from the 1990s and early 2000s, Pillay used the machinery of model-theoretic binding groups to give a slick geometric account and generalization of Kolchin’s theory of strongly normal extensions and constrained cohomology. This series of two talks is intended to be expository, with its main goal being to introduce and frame the relevant model-theoretic notions of internality and binding groups within the context of differential algebra, as well as to go through Pillay’s argument that his generalized strongly normal extensions arise from logarithmic differential equations defined over algebraic *D*-groups.

# Differential fields — a model theorist’s view

In his book *Saturated Model Theory*, Gerald Sacks described differentially closed fields as “the least misleading” example of an Ω-stable theory. His remark was particularly prescient as many interesting model theoretic phenomena arise naturally in differential algebra. Model theory has been strangely effective in both solving and generating questions in differential algebraic geometry. I will survey some aspects of this interaction.

This talk is part of a weekend-long workshop in differential algebra. Details are available here.

# The Dixmier-Moeglin problem for D-varieties

By a D-variety we mean, following Buium, an algebraic variety V over an algebraically closed field k equipped with a regular section s: V–> TV to the tangent bundle of V. (This is equivalent to the category of finite dimensional differential-algebraic varieties over the constants.) There are natural notions of D-rational map and D-subvariety. Motivated by problems in noncommutative algebra we are lead to ask under what conditions (V,s) has a maximum proper D-subvariety over k. (Model-theoretically this asks when the generic type is isolated.) A necessary condition is that (V,s) does not admit a nonconstant D-constant, that is, a D-rational map from (V,s) to the affine line equipped with the

zero section. When is this condition sufficient? I will discuss this rather open-ended problem, including some known cases.

# Imaginaries in valued fields of positive characteristic

I will discuss the basic properties of the theory SCVF of separably closed valued fields, and the closely related theory SCVH of separably closed valued fields equipped with Hasse derivations. The main new result is elimination of imaginaries (in a suitable language). This is joint work with Martin Hils and Silvain Rideau.

# Trichotomy principle for partial differential fields

The Zilber trichotomy principle gives a precise sense in which the structure on a sufficiently well behaved one-dimensional set must have one of only three possible kinds: disintegrated (meaning that there may be some isolated correspondences, but nothing else), linear (basically coming from an abelian group with no extra structure), or algebro-geometric (essentially coming from an algebraically closed field). This principle is true in differentially closed fields when “one dimensional” is understood as “strongly minimal” (proven by Hrushovski and Sokolovic using the theory of Zariski geometries and then by Pillay and Ziegler using jet spaces).

When working with differentially closed fields with finitely many, but more than one, distinguished commuting derivations, there are sets which from a certain model theoretic point of view (having to do with the notion of a regular type) are one dimensional even though they are infinite dimensional from the point of view of differential dimension. Moosa, Pillay and Scanlon showed that a weakening of the trichotomy principle is true for these sets: if there is a counter example to the trichotomy principle, then one can be found for a set defined by linear PDEs.

In this lecture, I will explain in detail what the trichotomy principle means in differential algebra, how the reduction to the linear case works, and then how one might approach the open problems.

This is a joint event of the CUNY Logic Workshop and the Kolchin Seminar in Differential Algebra, as part of a KSDA weekend workshop.

# Classification of strongly normal extensions of a differential field, and related issues

The material is taken from a joint paper with M. Kamensky, “Interpretations and differential Galois extensions.” Given a differential field *K* with field of constants *k*, and a logarithmic differential equation over *K*, the strongly normal extensions of *K* for the equation correspond (up to isomorphism over *K*) with the connected components of *G(k)* where *G* is the Galois groupoid of the equation. This generalizes to other contexts (parameterized theory,….), and is also the main tool in existence theorems for strongly normal extensions with prescribed properties.

This is a joint event of the CUNY Logic Workshop and the Kolchin Seminar in Differential Algebra, as part of a KSDA weekend workshop.

# A Differential Hensel’s Lemma for Local Algebras

We will discuss a differential version of the classical Hensel’s lemma on lifting solutions from the residue field (working on a local artinian differential algebra over a differentially closed field). We will also talk about some generalizations; for example, one can remove the locality hypothesis by assuming finite dimensionality. If time permits, I will give an easy application on extensions of generalized Hasse-Schmidt operators. This is joint work with Rahim Moosa.

# A Differential Algebra Sampler

We discuss several problems involving differential algebraic varieties and ideals in differential polynomial rings. The first one is the completeness of projective differential varieties. We consider examples showing the failure to generalize (even in the finite-rank case) of the classical “fundamental theorem of elimination theory”. We also treat identification of complete differential varieties and a connection to the differential catenary problem. We finish by examining what proof-theoretic techniques have to say about the constructive content of results such as the Ritt-Raudenbush basis theorem and differential Nullstellensatz. Our remarks include work with James Freitag and Omar León-Sánchez as well as ongoing work with Henry Towsner.

# On the Existence of Differential Chow Varieties

Chow varieties are a parameter space for cycles of a given variety of a given codimension and degree. We construct their analog for differential algebraic varieties with differential algebraic subvarieties, answering a question of Gao, Li and Yuan. The proof uses the construction of classical algebro-geometric Chow varieties, the model theory of differential fields, the theory of characteristic sets of differential varieties, the theory of prolongation spaces, and the theory of differential Chow forms. This is joint work with Wei Li and Tom Scanlon.

# Applications of Differential Algebra to Algebraic Independence of Arithmetic Functions

We generalize and unify the proofs of several results of algebraic independence of arithmetic functions using a theorem of Ax on differential Schanuel conjecture. Along the way of we investigation, we found counter-examples to some results in the literature.

# Differential-Henselian Fields

I will discuss valued differential fields. What parts of valuation theory go through for these objects, under what conditions? Is there a good differential analogue of Hensel’s Lemma? Is there a reasonable notion of differential-henselization? What about differential-henselianity for systems of algebraic differential equations in several unknowns?

I will mention results as well as open questions. The results are part of joint work with Matthias Aschenbrenner and Joris van der Hoeven, and have turned out to be useful in the model theory of the valued differential field of transseries.

# Differential varieties with only algebraic images

Consider the following condition on a finite-dimensional differential-algebraic variety *X*: whenever *X→Y* is a dominant morphism, and dim(*Y*) < dim(*X*), then *Y* is (a finite cover of) an algebraic variety in the constants. This property is a specialisation to differentially closed fields of a model-theoretic condition that itself arose as an abstraction from complex analytic geometry. Non-algebraic examples can be found among differential algebraic subgroups of simple abelian varieties. I will give a characterisation of this property that involves differential analogues of “algebraic reduction” and “descent”. This is joint work with Anand Pillay.

# Interpretations and differential Galois extensions

We prove a number of results around finding strongly normal extensions of a differential field *K*, sometimes with prescribed properties, when the constants of *K* are not necessarily algebraically closed. The general yoga of interpretations and definable groupoids is used (in place of the Tannakian formalism in the linear case).

This is joint work with M. Kamensky.

# Geometrically Trivial Strongly Minimal Sets in DCF0

In this talk we look at the problem of describing the “finer” structure of geometrically trivial strongly minimal sets in **DCF**_{0}. In particular, I will talk about the ω-categoricity conjecture, recently disproved in its general form by James Freitag and Tom Scanlon, and the unimodularity conjecture, a weakening of the above conjecture and which came to life after the work on the second Painlevé equations.

# The general solution of a first order differential polynomial

This is the title of a 1976 paper by Richard Cohn in which he gives a purely algebraic proof of a theorem (proved analytically by Ritt) that gives a bound on the number of derivatives needed to find a basis for the radical ideal of the general solution of such a polynomial. I will show that the method introduced by Cohn can be used to give a modern proof of a theorem of Hamburger stating that a singular solution of such a polynomial is either an envelope of a set of solutions or embedded in an analytic family of solutions depending on whether or not it corresponds to an essential singular component of this polynomial. I will also discuss the relation between this phenomenon and the Low Power Theorem. This will be an elementary talk with all these terms and concepts defined and explained.

# On transformations in the Painlevé family

The Painlevé equations are nonlinear 2nd order ODE and come in six families *P _{1}–P_{6}*, where

*P*consists of the single equation

_{1}*y′′=6y*, and

^{2}+t*P*come with some complex parameters. They were discovered strictly for mathematical considerations at the beginning of the 20th century but have arisen in a variety of important physical applications. In this talk I will explain how one can use model theory to answer the question of whether there exist algebraic relations between solutions of different Painlevé equations from the families

_{2}–P_{6}*P*.

_{1}–P_{6}# Model theory and the Painlevé equations

The Painlevé equations are nonlinear 2nd order ODEs and come in six families P1,…, P6, where P1 consists of the single equation $y′′=6y^2+t$, and P2,…, P6 come with some complex parameters. They were discovered strictly for mathematical considerations at the beginning of the 20th century but have arisen in a variety of important physical applications, including for example random matrix theory and general relativity.

# Effective Bounds For Finite Differential-Algebraic Varieties (Part I)

Given a differential algebraic variety over a partial differential field, can one give bounds for the degree of the Zariski closure which depend only on the order and degree of the differential polynomials (but not the parameters) which determine the variety? We will discuss the general theory of prolongations of differential algebraic varieties as developed by Moosa and Scanlon, and use this theory to reduce the problem to a combinatorial problem (which will be discussed in detail in the second part of the talk). Along the way we will give numerous examples of the usefulness of the result, some of an arithmetic flavor. We will also describe some other applications of the theory of prolongations.

This is joint work with Omar Sanchez.