Kolchin Seminar in Differential Algebra
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Academic year 2005–2006

Last updated on May 14, 2017.
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Fridays, September 9 and 16, 2005

Richard Churchill, Hunter College and the Graduate Center
An algorithm for finding closed-form solutions of
linear homogeneous differential equations - an overview.

This is an introductory talk on the Galois theory of linear ordinary differential equations, aimed at students. The group of an equation will be defined, the type of information about solutions provided by the group will be illustrated, and methods of calculation will be indicated.

For lecture notes (corrected) click here

Friday, September 23, 2005

Jerry Kovacic, CCNY
An algorithm for solving second order linear homogeneous differential equations

The Galois group tells us a lot about a linear homogeneous differential equation - whether or not it has "closed-form" solutions. Using it, we have been able to develop an algorithm for finding "closed-form" solutions. First we will compute the Galois group of some very simple equations. We then will solve a more complicated one, using the techniques of the algorithm. This example illustrates how the algorithm was discovered and the kinds of calculations used by it.

For lecture notes click here.       For a scan of the 1986 paper click here.

Fridays, September 30 and October 7, 2005

Jerry Kovacic, CCNY
Picard-Vessiot theory, algebraic groups and group schemes

We start with the classical definition of Picard-Vessiot extension, and show that the Galois group is isomorphic to an algebraic subgroup of  GL(n). Next we introduce the notion of Picard-Vessiot ring and describe the Galois group as spec of a certain subring of a tensor product. We shall also show existence and uniqueness of Picard-Vessiot extensions, using properties of the tensor product. Finally we hint at an extension of the Picard-Vessiot theory by looking at the example of the Weierstraß \wp-function. We use only the most elementary properties of tensor products, spec, etc. We will define these notions and develop what we need. No prior knowledge is assumed.

For lecture notes click here.

Fridays, October 14 and 28, 2005

William Sit, CCNY
Introduction to computational differential algebra

Under both Ritt and Kolchin, basic differential algebra was developed from a constructive view point and the foundation they built has been advanced and extended to become applicable in symbolic computation. In the first talk, we begin with a study of the division algorithm and how it may be modified and used to perform reductions in polynomial rings, ordinary and then partial differential polynomial rings. The abstract notions of (partial) differential rings, fields, and differential polynomials will be covered and no prerequisite is necessary. We will use examples to illustrate how differential polynomials may be ordered and manipulated algebraically using Euclidean-like division. The goal is to apply the reduction algorithms for ideal membership decisions, when possible, and to "simplify" a given system of algebraic differential equations like reducing the order, degree, and number of unknowns, or breaking the system up into "simpler" systems. We will compare this with the analogous operations on algebraic systems. More formally, we will cover the concepts of term-ordering and ranking, partial reduction and reduction, autoreduced sets, Grobner basis and characteristic sets.

In the second talk, we will discuss methods to compute Grobner basis and characteristic sets and the role they play in computational differential algebra: Are there constructive methods, though not necessarily efficient, to solve basic decidability problems? Each algebraic problem has a differential version simply by adding the word "differential" to appropriate places. The ideal membership problem, "Can we tell if a given (differential) polynomial ideal contains a given (differential) polynomial?" will be revisited. Other questions to be discussed are: Can we tell if a given (differential) polynomial ideal is prime? or radical? Does a (differential) polynomial ideal have a finite basis? How do the algebraic and differential analogs differ?

These talks will be informal and aimed at beginning graduate students. More rigorous treatment is available from my tutorial paper.

For lecture notes of both talks click here

Friday, October 21, 2005

Sally Morrison, Bucknell
New Methods in the Computation of Differential Ideals

This talk is an introduction to my Satuday, October 22, talk at Hunter College. Click here for details. It is aimed at beginning graduate students.

For lecture notes click here.

Fridays, November 4, 18, December 2 and 9, 2005

Phyllis Cassidy, Smith College and CCNY
Four talks on affine differential algebraic groups

We have heard talks on Picard-Vessiot theory, and, the Ritt-Kolchin elimination theory of the differential polynomial ring. We now move to differential algebraic geometry, which Alexandru Buium, in his book, Differential algebra and Diophantine Geometry, calls a "new geometry", and, to differential algebraic groups, the group objects of the new geometry. In the affine case, the objects of differential algebraic geometry are the sets of zeros of differential polynomial ideals. The symmetries present in the defining differential polynomial ideals of groups and their homogeneous spaces, makes these ideals much more tractable. In particular, characteristic sets are much easier to compute. Let   \partial  be a set of  m  commuting derivation operators on a differentially closed differential field of characteristic 0. My talks will be a narrative of the present theory of affine differential algebraic groups, with emphasis on linear groups, the Galois groups of parametrized Picard-Vessiot theory. Our point of view will be classical. In particular, linear differential algebraic groups will be subgroups of  GL(n).

For abstracts of the individual talks click here.      For lecture notes of all four talks click here.

Friday, November 11, 2005

Bernard Malgrange, Universite Joseph Fourier, Grenoble
Formal reduction of differential equations

Friday, December 16, 2005

Adam Crock, Graduate Center
An introduction to schemes

For lecture notes click here.

Friday, January 27, 2006

Richard Cohn, Rutgers University, New Brunswick, NJ
The trouble with differential ideals

Fridays, February 3 and February 10, 2006

Michael Tepper, The Graduate Center
Heights on varieties and the canonical height function,
Part 1: Number fields, Part 2: Function fields

I will develop the height function and canonical height function for points on varieties over number fields and over function fields. Then I will describe the basic finiteness property for points of bounded height, which holds for the number field case but not for the function field case.

For lecture notes of part 1 click here       For lecture notes of part 2 click here

Friday, February 24, 2006

Adam Crock, The Graduate Center
A different view of Groebner bases

Friday, March 10, 2006

Masood Aryapoor, Yale University
A short introduction to sheaf cohomology

Friday, March 17, 2006

Adam Crock, The Graduate Center
A different view of Groebner bases, continued

Friday, March 24, 2006

Peter Landesman, The Graduate Center
The connected component of a differential algebraic group is absolutely irreducible

Friday, April 7, 2006

Eugueny Pankratiev, Moscow State University
Standard differential bases

For lecture notes click here.

Friday, April 28, 2006

William Sit, CCNY
Computations and some open problems

I'll discuss some open problems in differential algebra for which experiments in computations may be helpful. Examples showing how to set up such experiments will be given.

For lecture notes click here.

Friday, May 5, 2006

Lourdes Juan, Texas Tech University, Lubbock
Torsors and Galois cohomology

This talk will contain background material for the Saturday talk. We will explain how the Galois cohomology with coefficients in a linear algebraic group  G  classifies the isomorphism classes of  G  torsors as well as the connection with Picard-Vessiot  G -extensions.

For lecture notes click here.

Friday, May 12, 2006

Peter Landesman, The Graduate Center
Classification of generalized  Gm-extensions

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