# Blog Archives

# Topic Archive: differential algebra

# 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.

Tuesday, October 4, 2016

# DART VII

DART VII, the seventh meeting of the conference series Differential Algebra and Related Topics, will take place at the CUNY Graduate Center (and other CUNY venues) from September 30 – October 4, 2016. Information about this conference is available here. It will include talks by Alexander Buium, Zoe Chatzidakis, Taylor Dupuy, James Freitag, Victor Kac, Joel Nagloo, Anand Pillay, and Thomas Scanlon, among others.

# DART VII

DART VII, the seventh meeting of the conference series Differential Algebra and Related Topics, will take place at the CUNY Graduate Center from September 30 – October 4, 2016. Information about this conference is available here.

# A model theory of affine n-space via differential algebra

Affine $n$-space $A_k^n$ and its algebraic equivalent, the polynomial ring $k[x_1,…,x_n]$, are basic and widely studied objects in geometry and algebra, about which we know a great deal. However, there remains a host of basic open problems (like the Jacobian conjecture, Zariski Cancellation Conjecture, Complement Problem, …) indicating that our knowledge is nonetheless quite limited. In fact, the greatest obstacle in solving the above conjectures is our inability to “pinpoint” affine space among all varieties (or $k[x]$ among all finitely generated $k$-algebras): this is the so-called Characterization Problem.

The most recent approach to these problems is via additive group actions on affine $n$-space, which corresponds on the algebraic side, to the theory of locally nilpotent derivations. Using this, for instance, N. Gupta recently showed the falsitude of the Zariski Cancellation Conjecture in positive characteristic.

From a model-theoretic point of view, the polynomial ring (in its natural ring language) is quite expressive: in characteristic zero, one can define the integers (as a subset), one can express in general that, say, Embedded Resolution of Singularities holds, etc. Of course, one of the peculiarities of model theory (and probably one of the reasons for its pariah status) is the unavoidable presence of non-standard models. In other words, a characterization problem is never solvable in model theory, unless one allows some non first-order conditions as well (e.g., cardinality in categorical theories–but most mainstream mathematicians would not be too happy about that either). But other, more intrinsic problems arise: there are elementary equivalent fields whose polynomial rings are not. So, can we find an expanded language plus a “natural” but non first-order condition, that pinpoints the standard model, i.e., $k[x]$ within the models of its theory. Or even better, since these complete theories will have unwieldy axiomatizations, can we find a (recursive?) theory, whose only model satisfying the extra non first-order condition is $k[x]$?

In view of the recent developments in algebra/geometry, to this end, I will propose in this talk some languages that include additional sorts, in particularly, a sort for derivations. This is different from the usual language of differential fields, where one only studies a fixed (or possibly finitely many) derivation: we need all of them! We also need a substitute for the notion of degree, and the corresponding group $Z$-action as power maps. To test our theories, we should verify which algebraic/geometric properties are reflected in this setup. For instance, affine $n$-space has no cohomology, which is equivalent to the exactness of the de Rham complex, and this latter statement is true in any of the proposed models. Nonetheless, this is only a preliminary analysis of the problem, and nothing too deep will yet be discussed in this talk.

# 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.

# 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.

# On the Existence of Parametrized Strongly Normal Extensions

In this talk we look at the problem of existence of differential Galois extensions for parameterized logarithmic equations. More precisely, if E and D are two distinguished sets of derivations and K is an E union D-field of characteristic zero, we look at conditions on (K^E,D), the E-constants of K, that guarantee that every (parameterized) E-logarithmic equation over K has a parameterized strongly normal extension. This is joint work with Omar Leon Sanchez.

# Effective bounds for the existence of differential field extensions

We present a new upper bound for the existence of a differential field extension of a differential field

(K; D) that is compatible with a given field extension of K. In 2014, Pierce provided an upper bound in

terms of lengths of certain antichain sequences of N^m equipped with the product order. Pierce’s theory

has interesting applications to the model theory of fields with m commuting derivations, and his results

have been used when studying effective methods in differential algebra, such as the effective differential

Nullstellensatz problem. We use a new approach involving Macaulay’s theorem on the Hilbert function

to produce an improved upper bound. In particular, we see markedly improved results in the case of two

and three derivations.

This is joint work with Omar Leon Sanchez.

# 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.