**The Kolchin Seminar in Differential
Algebra **

**Saturday March 19 ^{th}
2005**

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**Jerry Kovacic**City College - CUNY

2:30 - 4:00PM

**Abstract:** There are many equivalent definitions of a simple or
non-singular point in algebraic geometry. They involve the Jacobi condition on
the rank of a matrix, the dimension of m/m^2, the minimal number of generators
of m, etc. All of these rely on the various notions of dimension: Krull,
topological, transcendence degree, etc. In differential algebra there is no good
notion of dimension so the it is unclear what the "correct" definition of simple
point should be. In this very preliminary talk we investigate various possible
definitions. We start by looking at the tangent space and cone, a la Mumford.
The equality of these is probably not sufficient as examples show. We then
consider the intersection of powers of the maximal ideal. In contrast to what
happens in algebraic geometry, this intersection need not be (0). The best
definition seems to be the one due to Johnson. But it is the worst in that it
depends on a choice of generators. Using a characteristic set we can get
something vaguely analogous to the Jacobi condition. However it depends on the
choice of ranking. We also look at a Krull dimension polynomial, and numerical
polynomials for the dimension of m/m^2, number of generators for m, etc.
Unfortunately the obvious inequalities between these go the wrong way! We end by
giving up, but hope that the audience has some fruitful ideas.

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**Phyllis Cassidy**

City College -
CUNY**Toward a new approach to linear differential algebraic
groups**

4:30 - 6:00 PM

**Abstract:** After describing recent work with Michael F. Singer on the
beginnings of a Galois theory of parametric linear differential equations, based
on the approach to Picard-Vessiot theory of Levelt, van der Put-Singer, and
Kovacic, I will indicate how this formulation requires a more Tannakian approach
to the theory of linear differential algebraic groups.

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**Coffee and tea will be served beginning at 2:00 PM Room
HW706 **

**Click here for directions
to Hunter College and location of room**