Kolchin Seminar in Differential Algebra 
 The Graduate Center 365 Fifth Avenue, New York, NY 100164309 General Telephone: 12128177000 
Academic
year 2005–2006 Last updated on January 18, 2018. 
Fridays, September 9 and 16, 2005
Richard Churchill, Hunter College and the Graduate Center
An algorithm for finding closedform 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.
Friday, September 23, 2005
Jerry Kovacic, CCNY
An algorithm for solving second order linear homogeneous differential equationsThe Galois group tells us a lot about a linear homogeneous differential equation  whether or not it has "closedform" solutions. Using it, we have been able to develop an algorithm for finding "closedform" 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, CCNYFridays, October 14 and 28, 2005
PicardVessiot theory, algebraic groups and group schemesWe start with the classical definition of PicardVessiot extension, and show that the Galois group is isomorphic to an algebraic subgroup of GL(n). Next we introduce the notion of PicardVessiot ring and describe the Galois group as spec of a certain subring of a tensor product. We shall also show existence and uniqueness of PicardVessiot extensions, using properties of the tensor product. Finally we hint at an extension of the PicardVessiot theory by looking at the example of the Weierstraß 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.
William Sit, CCNY
Introduction to computational differential algebraUnder 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 Euclideanlike 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 termordering 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.
Friday, October 21, 2005
Sally Morrison, Bucknell
New Methods in the Computation of Differential IdealsThis talk is an introduction to my Satuday, October 22, talk at Hunter College. Click here for details. It is aimed at beginning graduate students.
Fridays, November 4, 18, December 2 and 9, 2005
Phyllis Cassidy, Smith College and CCNY
Four talks on affine differential algebraic groupsWe have heard talks on PicardVessiot theory, and, the RittKolchin 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 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 PicardVessiot 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
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 fieldsI 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
Friday, April 28, 2006
William Sit, CCNY
Computations and some open problemsI'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.
Friday, May 5, 2006
Lourdes Juan, Texas Tech University, Lubbock
Torsors and Galois cohomologyThis 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 PicardVessiot G extensions.
Friday, May 12, 2006
Peter Landesman, The Graduate Center
Classification of generalized G_{m}extensions
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