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The Kolchin Seminar in Differential Algebra

Hunter College - Room HW706

Saturday September 13th 2003

Marius van der Put
Groningen University, The Netherlands
Moduli space of differential equations, I
2:00 - 3:00PM

Abstract:   One considers linear differential equations on the projective line (or on the complex sphere) with prescribed singularities at prescribed points. It turns out that a fine moduli space and a universal family exist. A rather special case, namely a given irregular singularity at the origin and a regular singularity at the infinite point, leads to a better understanding of the classification of meromorphic differential equations (as given by J.P. Ramis, J. Martinet, D.G. Babbitt, V.S. Varadarajan and others). The concept of fine moduli spaces and more general families of linear differential equations on the complex sphere seems to be the proper setting for M.F. Singer's (old) results on the behaviour of the differential Galois group in a family of equations. In particular, we will sketch a proof of his ``constructibility theorem'' in this new setting. There are also relations with braid groups and Painlevé equations.

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Jerry Kovacic
City College - CUNY
The differential Galois theory of strongly normal extensions
3:30 - 4:30PM

Abstract: Differential Galois theory, the theory of strongly normal extensions, has unfortunately languished, possibly due to its reliance on Kolchin's elegant, but not widely adopted, axiomatization of the theory of algebraic groups. Here we attempt to revive the theory using affine differential schemes. We also avoid Weil's ``group chunks'' and get the Galois group canonically identified (not merely birationally) with the closed points of an affine differential scheme. We do not need a universal differential field, but use a certain tensor product instead. This tensor product is a Sweedler coring in a natural way and the comultiplication translates into the Galois group operation. Diffspec of this tensor product splits, i.e. is obtained by base extension from a (not differential, not necessarily affine) group scheme. And the converse also holds. If diffspec of a certain tensor product splits then it comes from a strongly normal extension. Thus we have a geometric characterization of the notion of strongly normal extension.

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Marius van der Put
Groningen University, The Netherlands
Moduli space of differential equations, II
5:00 - 6:00 PM

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

Click here for directions to Hunter College and location of room

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