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.

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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.

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Rutgers
University

**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*; *2*1…*)
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.

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

For
further information, please visit

**http://math.hunter.cuny.edu/ksda**