Kolchin Seminar in Differential Algebra

Academic year 2006–2007 Last updated on January 31, 2020. Other years: 2005–2006   2007–2008   2008–2009   2009–2010   2010–2011   2011–2012   2012–2013   2013–2014   2014–2015   2015–2016   2016–2017   2017–2018   2018–2019   2019–Fall

In the fall we concentrated on differential Galois (Picard-Vessiot) theory.     For a general reference click here.

September 1, 2006

Richard Churchill, Hunter College and The Graduate Center, CUNY
Why we cannot integrate  ∫ exp(x²) dx  in elementary calculus

For lecture notes click here.

September 8, 2006

Richard Churchill, Hunter College and The Graduate Center, CUNY
Introduction to the Galois theory of linear ordinary differential equations

For lecture notes click here

September 15, 2006

Adam Crock, The Graduate Center
A brief introduction to differential algebra

This is the first in a series of lectures on differential Galois theory. This lecture will introduce the basics of differential rings, ideals, homomorphisms, polynomials and extensions. These concepts and the analogous concepts of commutative algebra will be contrasted. This is an introductory lecture: only elementary abstract algebra is assumed.

For more details about differential rings click here

September 29, 2006

Adam Crock, The Graduate Center
Introduction to differential algebra, part 2

This is the second in a series of lectures on differential Galois theory. We will discuss radical, prime and maximal differential ideals. The notions of differentially algebraic and differentially transcendental will also be introduced.

October 6, 2006

Adam Crock, The Graduate Center
Introduction to differential algebra, part 3

This is the third in a series of lectures on differential Galois theory. The topics include differential polynomials and the Wronskian.

October 13, 2006

Richard Churchill, Hunter College and The Graduate Center
A geometric approach to linear ordinary differential equations

We offer a formulation of linear ordinary differential equations midway between what one encounters in a first undergraduate ODE course and what one encounteres in a graduate Differential Geometry course (in the latter instance under the heading of "connections"). The talk should be accessible to first-year graduate students. Analogies with elementary linear algebra are emphasized; no familiarity with Differential Geometry is assumed.

For lecture notes click here (revised).

October 20, 2006

Richard Churchill, Hunter College and The Graduate Center
A geometric approach to linear ordinary differential equations, Part 2

A Picard-Vessiot extension of a linear ODE is the differential algebraic analogue of the splitting field of a polynomial in ordinary Galois theory. I will give the definition and prove uniqueness.

For lecture notes click here (revised).

Jerald Kovacic, City College
Existence of a Picard-Vessiot extension for a differential module

We will start the proof that a Picard-Vessiot extension for a given differential module (or equivalently a linear homogeneous differential equation) exists. We will use tensor products and differentially simple differential rings.

For lecture notes click here.

October 27, 2006

Jerald Kovacic, City College
Existence of a Picard-Vessiot extension

We will continue the proof of the existence of a Picard-Vessiot extension, after a review of the short talk presented the previous week.

For lecture notes click here.

November 3, 2006

Jerald Kovacic, City College
The Picard-Vessiot ring

We will continue the development of the Picard-Vessiot theory.

For lecture notes click here.

November 10, 2006

Bernard Malgrange, Institut Fourier, Université de Grenoble
Lie pseudo groups

November 17, December 1, December 8, 2006

Phyllis J. Cassidy, Smith College and City College
The Galois group of a Picard-Vessiot extension

For November 17 lecture notes click here.
For December 1 lecture notes click here.
For December 8 lecture notes click here.

December 8, 2006

Alexander A. Mikhalev, Moscow State University and CCNY
Differential Lie superalgebras

Differential Lie algebras appeared in papers of R. Baer (1927) and N. Jacobson (1937) devoted to derivations of associative rings. V. K. Kharchenko used them (1979) to study differential identities of semiprime rings. General theory of differential Lie algebras was exposed in the monographs:

K. I. Beidar, W. Martindale and A. V. Mikhalev, Rings with Generalized Identities, Marcel Dekker, 1996.
V. K. Kharchenko, Derivations of Associative Rings, Kluwer, 1991.

We consider differential Lie superalgebras and prove analogs of Poincare-Birkhoff-Witt theorem for universal enveloping algebras of these algebras. In the proof we use the theory of Groebner-Shirshov bases of ideals of free algebras.

December 15, 2006

Evelyne Hubert, INRIA, presently at IMA, Minneapolis.
Algebra and algorithms for differential elimination and completion

Differential algebra provides an algebraic viewpoint on nonlinear differential systems. The motivating questions for this talk are:

* How do we define the general solution of a nonlinear equation?
* What are the conditions for a differential system to have a solution?
* How do we measure the "degrees of freedom" for the solution set of a differential system?

The theory and algorithms for those are extensions of commutative algebra (prime ideal decomposition, Hilbert polynomials) and Groebner bases techniques.

The library diffalg in Maple supports this introduction to constructive differential algebra. It has been developed by F. Boulier (1996) and the speaker afterwards. A recent extension of differential algebra to non-commutative derivations, and its implemenation in diffalg, allows us to treat systems bearing on differential invariants.

For lecture notes click here.

February 2, February 9, 2007

Christopher Seaman, The Graduate Center
An introduction to Grobner bases

February 16, February 23, March 2, 2007

Adam Crock, The Graduate Center
Rankings and characteristic sets

March 9, 2007

William Sit, The City College
Elimination Term-Orderings

Elimination of variables in a system of algebraic polynomials can be done with Grobner basis computations by suitably choosing a total order for the set of monomials compatible with multiplications. This talk will give the basic theory, with examples.

March 16, March 23, March 30, 2007

Russell Miller, Queens College
Computable Model Theory and Differential Algebra

Model theory is the study of mathematical structures and the extent to which they can be described by statements and formulas. Computable model theory considers the effectiveness of results in model theory: whether they can actually be given or realized by algorithms. For example, a computable field is a field  F  in which the basic operations of addition and multiplication can be computed algorithmically, and one can then ask whether there exists a splitting algorithm for deciding whether a given polynomial in  F[X₁,...,X_n]  is reducible there.

We will give a tutorial in computable model theory, oriented towards results on fields and towards an audience with no serious background in either computability or model theory. Differential algebra is a natural subject for study by computable model theorists, yet there are precious few results for computable differential fields. (It should be understood that this is not the same thing as computational differential algebra, although there certainly should be some relation between the two.) As an example, we will describe Rabin's well-known result that every computable field  F  has a computable algebraic closure, but that  F  itself can be a computable subfield of the algebraic closure iff there is a splitting algorithm for  F[X]. One would expect some sort of analogous result for computable differential fields and their differential closures, yet to the speaker's knowledge, no such work has been done.

Computable model theory has always restricted itself to countable structures, since the natural domain for computability is the natural numbers. However, we will present work by the speaker which also considers certain uncountable structures  S, called locally computable structures, by effectively describing the finitely generated substructures of  S, rather than giving a global description of  S. This concept was only recently developed and has not as yet been widely applied, but fields and differential fields are natural choices for its use.

Friday, April 20 and April 27, 2007, 11AM

Anupam Bhatnagar, The Graduate Center
Algebraic Geometry on Difference Rings

We will discuss some ideas from Hrushovski's paper: "Elementary Theory of Frobenius Automorphisms".

Friday, May 4, 2007

Daniel Pasca, Hunter College
Differential algebraic techniques in Hamiltonian Dynamics

We show how differential algebraic techniques can be used to establish the non-integrability of complex analytic Hamiltonian systems.

Friday, May 11, 2007

Anupam Bhatnagar, The Graduate Center
Algebraic Geometry on Difference Rings

We will discuss some ideas from Hrushovski's paper: "Elementary Theory of Frobenius Automorphisms".

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