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The Kolchin Seminar in Differential Algebra

Hunter College - Room HW706

Saturday - Sunday December 14-15 2002

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Saturday Session

Michèle Audin
Université Louis Pasteur and IRMA, Strasbourg, France
Real non-integrability results: the example of the satellite, I
2:00 - 2:45 PM

Michèle Audin
Université Louis Pasteur and IRMA, Strasbourg, France
"Real non-integrability results: the example of the satellite, II
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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Sunday Session

Lourdes Juan
Texas Tech University
Generic Picard-Vessiot Extensions and generic linear differential equations with group G
10:30 - 11:30 AM

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

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