**The
Kolchin Seminar in Differential Algebra**

**Saturday
September 18**^{th}

**John
Nahay****
**Swan Orchestral Systems

**Abstract:**

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**Phyllis Cassidy**

**Parametrized Picard-Vessiot extensions**

**Abstract:** These talks describe a new research project, joint
work with Michael F. Singer. Consider the Kummer equation

.

It is a second order
differential equation on the complex plane, in the operator , with coefficients rational in , and, in a parameter . The equation has a
regular singularity at , and, an irregular singularity at . The special function , known as the lower incomplete gamma function, is a solution. Let
be a of . If we adjoin and all its and, derivatives to , we will have an example of a parametrized Picard-Vessiot
extension with Galois group the additive group . If, we then, adjoin , the Galois group of the extension is
the multiplicative group If we descend to , the differential extension
field generated by has Galois group the semi direct
product , where is the subgroup of defined by the
differential equation (equivalently, ). We should note that, although
has finite absolute dimension (it depends on arbitrary constants), it can not be realized as an algebraic
subgroup of , for any positive integer (it does not descend to constants). is, itself,
the Galois group of the intermediate differential field over . The fact that does not
descend to constants tells us that the parametrized Picard-Vessiot extension of is not a classical Picard-Vessiot
extension. It is, however, a classical Picard-Vessiot extension of . It is time to step
back, and, move from these interesting examples to the general theory of
parametrized Picard-Vessiot extensions. In these talks, I show how the theory
has a natural context: the ``new'' geometry called *differential algebraic
geometry*. Since the Galois groups are linear, we remain comfortably in affine
differential algebraic geometry. We will also see how differential closures,
developed by Shelah, Kolchin, McGrail, and others, are efficient replacements
here for Kolchin's universal differential fields. We have three sets of
derivation operators to keep track of: The ``Galois theory'' derivation , the set of derivations , with respect to
the parameters, and, the union of the two sets of
derivations. The field of constants of , consisting of functions of the parameters,
is both a -field and a -field. The interplay of -closures and -closures is very
interesting. Parametrized Picard-Vessiot extensions are a special case of Peter
Landesman's new theory, *generalized strongly normal extensions*, in
which, the Galois groups are differential algebraic groups. Landesman's theory
is based on Kolchin's axiomatic treatment of differential algebraic groups.
Since the Galois groups of parametrized Picard-Vessiot extensions are linear
groups, the foundations of the theory can be developed easily *ab initio*.
Moreover, the theory is a striking example of the role that differential
algebraic groups play as symmetry groups of systems of polynomial differential
equations. The symmetry action here is the right regular representation, which
enables us to prove easily the Fundamental Theorem of Galois theory, even in
the case where the set of *isomorphisms*, rather than the group of
automorphisms, is given the structure of differential algebraic group, and, no
conditions are placed on the ground differential field. This enables us to keep
track of the dependence of the elements of the extension field on the
parameters, as well as on . In this generalization of Picard-Vessiot
theory, differential algebraic groups are realized as groups of deformations of
systems of differential equations that depend on parameters. The special
function known as the incomplete gamma function provides us with a realization
of a differential algebraic group of infinite absolute dimension (it depends on
infinitely many arbitrary constants) as a Galois group. Its derivative, , with respect to , gives us an
example of a parametrized Picard-Vessiot extension of *finite *transcendence
degree, which, nevertheless, is *not* a classical Picard-Vessiot
extension.

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**Phyllis Cassidy**

**Parametrized Picard-Vessiot extensions (continued)**

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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**