**The Kolchin Seminar in Differential Algebra
**

**Saturday September
13**^{th}** 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

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

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
**