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The Kolchin Seminar in Differential Algebra

Hunter College - Room HW706

Saturday March 19th 2005

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Jerry Kovacic
City College - CUNY
Some thoughts on simple points in differential algebraic geometry
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

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