The Kolchin Seminar in Differential Algebra

Saturday - Sunday December 14-15 2002

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__Saturday Session__

** Michèle Audin
**
Université Louis Pasteur and IRMA, Strasbourg, France

2:00 - 2:45 PM

** Michèle Audin
**
Université Louis Pasteur and IRMA, Strasbourg, France

3:00 - 3:45 PM

**Abstract:** After a short presentation of what an integrable system is and why it is an
interesting notion, I will show how techniques of symplectic and real algebraic
geometry (together with the Morales-Ramis theorem) can be used to prove that a
Hamiltonian system is non-integrable. I will apply this to the system describing
the "attitude" of a satellite around the Earth.

** Claude Mitschi**

Université Louis Pasteur, Strasbourg, France

**Solvable-by-finite differential Galois groups**

(Joint work with Michael Singer)

4:00 - 5:00 PM

**Abstract:** I will present the following result: let G be a linear algebraic
group over an algebraic closed field C of characteristic zero, with a solvable
identity component, and let k=C(x). For any k-irreducible principal homogeneous
space V for G the derivation d/dx of k extends on k(V) in such a way that the
function field k(V) becomes a Picard-Vessiot extension of k with Galois group
G. The proof is constructive up to the embedding problem of classical Galois
theory, and it is in the spirit of Kovacic's pioneering work in this field.

**Mark van Hoeij **

University of Florida

**Descent for differential modules and skew fields**

5:15 - 6:15 PM

**Abstract:** We will study rationality questions for differential modules
and differential operators. If a differential operator L is equivalent to its
conjugates over k, is it then equivalent to an operator defined over k? We will
show how counter examples to this question correspond to skew fields, and will
make this correspondence explicit in both directions. Similar questions are
studied for projective equivalence of differential operators. The main tool is
the study of differential modules over skew fields.
** **

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**Texas Tech University

**Abstract:** The notion of generic linear differential equation was introduced by Lawrence Goldman in his paper "Specialization and Picard-Vessiot Theory", where he constructed such equations for various groups. Recently, we have constructed generic Picard-Vessiot extensions for connected linear algebraic groups G - which include the ones for which Goldman constructed a generic equation. The specialization properties of our extension are similar to those of Goldman's equation, so the question remains whether a group which admits a generic equation will also admit a generic extension and vice versa. We will discuss this question in our talk.
** **

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Coffee and tea will be served beginning at 1:30 PM on Saturday, and at 10:00 AM on Sunday Room HW706
**

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