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The Kolchin Seminar in Differential Algebra

Hunter College - Room HW706

Saturday April 16th 2005

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Kurusch Ebrahimi-Fard
Physikalisches Institut der Universitaet Bonn
Algebraic Birkhoff decomposition in pQFT, and the diagrammatic Renormalization Group differential equation
3:30 - 4:30PM

Abstract: Connes-Kreimer uncovered the mathematical structure underlying renormalization in pQFT in terms of Hopf algebras of Feynman graphs. The process of renormalization - in short, subtraction of divergencies in a physical sound way - is captured by an algebraic Birkhoff decomposition of Feynman rules. We briefly review these recent findings, and then start to explore the renormalization group differential equations from a diagrammatic point of view.

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Raymond T. Hoobler
City College (CUNY)
Generators of D-Modules in positive characteristic
5:00-6:00PM

Abstract: Let $ R=k[X_{1},\ldots ,X_{n}]$ and let $ f$ be a polynomial in $ R$. Then the localization of $ R$ with respect to $ f$, $ R_{f}$, is a module over the ring of differential operators, $ D:=D_{R/k}$. If char($ k$)=0, then $ R_{f}$ is a module of finite length over $ D$. This leads to the Bernstein-Sato polynomial $ b_{f}\in k[s]$ and a differential operator $ Q(S)\in D[s]$ such that

$\displaystyle Q(s)\cdot f^{s+1} = b_{f}(s)\cdot f^{s-1,}
$

for every $ s$ (and so $ f^{s_0}\in D\cdot f^{s_0+1}$ if $ s_0$ is not a root of $ b_f (s)$), one of the most startling results of $ D$-module theory. Last summer the analogue of this result was found when char$ (k)\ne
0$, and this case is even more surprising. It turns out that for a regular ring $ R$ of finite type over a field of positive characteristic with a $ p$-base, $ R_{f}$ is generated as a $ D$-module by $ f^{-1}$! I will sketch the argument of Alvarez-Montaner, Blickle, and Lyubeznik demonstrating this result, show how characteristic p differential operators come into the picture, and give an example of this phenomenon.

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Coffee and tea will be served beginning at 3:00 PM
Room HW706

Click here for directions to Hunter College and location of room

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