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The Kolchin Seminar in Differential Algebra

Hunter College - Room HW706

Saturday December 11th 2004

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Richard Cohn
Rutgers University
A reinterpretation of the difference equation analogue of Ritt's low power condition
2:00 - 3:00PM

Abstract: The low power condition introduced by J.F.Ritt is a necessary and sufficient condition for {y} to be a component of {A}, where A is an algebraically irreducible differential polynomial in the polynomial differential ring K{y}. (In its more general form, for any prime differential ideal to be a component). Stated differently, failure of this condition is sufficient but not necessary for A to have a non-zero solution specializing to 0.
There is an analogous condition in difference algebra, the low weight condition. The condition is necessary but not sufficient for {y} to be a component of {B}, where B is an algebraically irreducible polynomial in the difference ring F{y}. Sufficiency holds if B is of second order and in other interesting cases, but fails at third order. Stated differently, failure of the low weight condition is sufficient but not necessary for B to have a non-zero solution specializing to 0.
A full understanding of when {y} is a component of {B} has escaped my investigations and appears difficult. Recently I have found a new interpretation of the low weight condition which sidesteps this difficulty. Failure of the low weight condition is necessary and sufficient for B to have a non-zero solution which specializes "nicely" to 0, where by "nicely" I mean a specialization that extends to a difference closed place, that is a place whose valuation ideal is closed under the difference operator. (The corresponding result is also valid in the differential case). Further development of this idea will probably require Barbara Lando's difference algebra analogue of Morrison theory of differential places.

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Li Guo
Rutgers Uiversity
From quantum field theory renormalization to differential Galois groups
3:30 - 4:30PM

Abstract: We will present the recent work of Connes and Marcolli that relates the Hopf algebra approach of Connes and Kreimer on QFT to differential Galois groups. We will also underline the key role play by Rota-Baxter algebras. The main references are

Title: From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory
Authors: Alain Connes (College de France), Matilde Marcolli (MPIM)
http://www.arxiv.org/abs/hep-th/0411114

Title: Renormalization and motivic Galois theory
Authors: Alain Connes (College de France), Matilde Marcolli (MPIM Bonn)
http://www.arxiv.org/abs/math.NT/0409306

Title: Spitzer's Identity and the Algebraic Birkhoff Decomposition in pQFT
Authors: Kurusch Ebrahimi-Fard, Li Guo, Dirk Kreimer
http://www.arxiv.org/abs/hep-th/0407082

Title: Integrable Renormalization II: the general case
Authors: Kurusch Ebrahimi-Fard, Li Guo, Dirk Kreimer
http://www.arxiv.org/abs/hep-th/0403118

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Alexey Ovchinnikov
North Carolina State University
Sections of a differential spectrum
5:00 - 6:00PM

Abstract: pdf

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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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