The Kolchin Seminar

Hunter West, Room HW706

Saturday, September 14th, 2002


Professor Sally Morrison
Bucknell University
“Differential Polynomial Algebra”
2:00 - 3:00 PM

Abstract. This is an introduction to the language and methods of Differential Algebra for those with little or no previous exposure to the subject. The emphasis will be on examples and major ideas rather than on precise techniques and proofs.

Differential polynomial algebra may be regarded as an attempt to use the methods of abstract algebra to obtain information about solutions of systems of differential equations, in much the same way that the theory of polynomial rings yields information about solutions of systems of polynomial equations.

In this talk we review the fundamental definitions and problems of the subject including: the basics of differential rings, ideals, and homomorphisms; the correspondence between solutions of differential equations and differential ideals; decomposition of a solution into its irreducible components—and, in particular, the decomposition of a single differential polynomial into its general and singular components; and the notion of differential dimension.



Professor Jerald J. Kovacic

City College, CUNY


3:30  - 4:30 PM

Abstract. The language of schemes, which has proven to be of value to algebraic geometry, has not yet been widely accepted into differential algebraic geometry. One reason may be that there are some “challenges”. In this talk we examine some of those challenges. The first work on diffspec of a differential ring is due to Keigher and this is the definition we use. It is different from those of Carra’ Ferro and Buium. We first examine the ring of global sections, which in general is not isomorphic to the given differential ring. This brings up the notions of differential zero, differential unit and AAD (Annihilators Are Differential). Morphisms of diffspec also pose challenges; they are not always induced from homomorphisms of differential rings. Products and closed subschemes present further challenges. We also introduce the associated space of constants of a differential scheme and the notion of split differential scheme. These ideas are used in differential Galois theory. We end with a challenge to the audience to develop theories of quasi-coherent sheaves, group schemes, dimension, singularities, etc.



Professor Earl J. Taft

Rutgers University

“Generating Combinatorial Identities from Solutions of

Difference Equations with Polynomial Coefficients”

5:00 - 6:00 pm

Abstract. We consider sequences s = (s_n), with coordinates in a field, which satisfy a difference operator, i.e., a polynomial f(D) in the difference operator D ( D is the shift to the left).If f(D) has constant coefficients, then s is a linearly recursive sequence. For example, the Fibonacci sequence (1; 1; 2; 3; 5; 8; 13; 21…) satisfies D^2 - D - I. More generally, we consider solutions of f(D) when f(D) has polynomial coefficients. A polynomial p(x) acts on sequences by p(x)s = (p(n)s_n). For example (n!) satisfies D -  (x + 1)I. A polynomially recursive sequences has a comultiplication (diagonalization). This can be interpreted as a combinatorial identity on the coordinates of the sequence. Examples of such identities will be given. We will concentrate on the algorithmic aspects involved in producing such identities, rather than on the theoretical algebraic and topological aspects of the space of polynomially recursive sequences.



Coffee and tea will be served beginning at 1:30 pm

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