Kolchin Seminar in Differential Algebra 

The Graduate Center 365 Fifth Avenue, New York, NY 100164309 General Telephone: 12128177000 
Academic year 2006–2007
Last updated on May 14, 2017. 
In the fall we concentrated on differential Galois (PicardVessiot) theory. For a general reference click here.
September 1, 2006
Richard Churchill, Hunter College and The Graduate Center, CUNY
Why we cannot integrate ∫ exp(x^{2}) dx in elementary calculus
September 8, 2006
Richard Churchill, Hunter College and The Graduate Center, CUNY
Introduction to the Galois theory of linear ordinary differential equations
September 15, 2006
Adam Crock, The Graduate Center
A brief introduction to differential algebraThis 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.
September 29, 2006
Adam Crock, The Graduate Center
Introduction to differential algebra, part 2This 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 3This 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 equationsWe 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 firstyear graduate students. Analogies with elementary linear algebra are emphasized; no familiarity with Differential Geometry is assumed.
October 20, 2006
Richard Churchill, Hunter College and The Graduate Center
A geometric approach to linear ordinary differential equations, Part 2A PicardVessiot 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 PicardVessiot extension for a differential moduleWe will start the proof that a PicardVessiot extension for a given differential module (or equivalently a linear homogeneous differential equation) exists. We will use tensor products and differentially simple differential rings.
October 27, 2006
Jerald Kovacic, City College
Existence of a PicardVessiot extensionWe will continue the proof of the existence of a PicardVessiot extension, after a review of the short talk presented the previous week.
November 3, 2006
Jerald Kovacic, City College
The PicardVessiot ringWe will continue the development of the PicardVessiot theory.
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 PicardVessiot extensionFor 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 superalgebrasDifferential 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.We consider differential Lie superalgebras and prove analogs of PoincareBirkhoffWitt theorem for universal enveloping algebras of these algebras. In the proof we use the theory of GroebnerShirshov bases of ideals of free algebras.
V. K. Kharchenko, Derivations of Associative Rings, Kluwer, 1991.
December 15, 2006
Evelyne Hubert, INRIA, presently at IMA, Minneapolis.
Algebra and algorithms for differential elimination and completionDifferential 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 noncommutative derivations, and its implemenation in diffalg, allows us to treat systems bearing on differential invariants.
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 TermOrderingsElimination 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 AlgebraModel 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[X_{1},...,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 wellknown 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 RingsWe 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 DynamicsWe show how differential algebraic techniques can be used to establish the nonintegrability of complex analytic Hamiltonian systems.
Friday, May 11, 2007
Anupam Bhatnagar, The Graduate Center
Algebraic Geometry on Difference RingsWe will discuss some ideas from Hrushovski's paper: "Elementary Theory of Frobenius Automorphisms".
2005–2006
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