Home

The Kolchin Seminar in Differential Algebra

Hunter College - Room HW706

Saturday May 15th 2004

******************************************

Andy R. Magid
University of Oklahoma
Iterated Picard-Vessiot extensions
2:00-3:00PM

Abstract: The consideration of solutions of differential equations whose coefficients are solutions of differential equations leads to finite towers of Picard-Vessiot (or Differential Galois) extensions of a differential field F. While such iterated Picard-Vessiot extensions may not themselves be even embeddable in a Picard-Vessiot extension, they do embedd in iterated Picard-Vessiot closures, and the automorphism groups of these latter, as we show in the main result, may be used to construct a Galois correspondence for the differential subfields of normal locally iterated Picard-Vessiot extensions. In the process, we characterize the differential subfields of the iterated closures.

******************************************

William Sit
City College of CUNY
Some symbolic computation software for differential equations
3:30-4:30PM

Abstract: A system of partial differential equations that occurs in scientific applications can usually be transformed to a system where the dependent variables and their derivatives appear polynomially in the equations. Using purely algebraic manipulations and differentiation, computer algebra systems such as Maple or Mathematica can be used to perform change of variables and (differential) eliminations of dependent variables that may transform the system into simpler forms such as lowering the orders or degrees of the equations. For partial differential equations, it is sometimes possible to derive the symmetries of the system or find some exact solutions.
This talk is a preliminary report of an attempt to survey a small portion of recent symbolic computational software of interest to differential algebraists and hopefully also to researchers looking for software for differential equations based on Maple, Mathematica and Axiom. A selected subset of software implementations will be illustrated with examples of systems of differential equations that benefited from these techniques.
The talk is intended for the general scientific public.

******************************************

Andy R. Magid
University of Oklahoma
The construction of Picard-Vessiot closures
5:00-6:00PM

Abstract: There are two basic approaches to the algebraic construction of Picard-Vessiot closures: one can either construct a maximal extension of a suitable sort by an application of Zorn's Lemma, and then try to prove that it contains copies of all Picard-Vessiot extensions of the base; or one can take a tensor product of all the Picard-Vessiot extensions of the base and then try to prove than an appropriate quotient exists. (Both approaches are related, of course.) We follow the second approach. Our argument proceeds via a class of differential fields which are especially well adapted for the Zorn's lemma argument we need.It is known that differential automorphisms of the base field lift to differential automorphisms of a Picard-Vessiot closure. We give another proof of that, using the tensor product construction of closures, which makes this lifting theorem more transparent.

******************************************

Coffee and tea will be served beginning at 1:30 PM
Room HW706

Click here for directions to Hunter College and location of room

Home