1 A compilation of two sets of notes at the University of Kansas; one in the Spring of 2002 by ?? and the other in the Spring of 2007 by Branden Stone. These notes have been typed

Fedosov has described a geometro-algebraic method to construct in a canonical way a deformation of the Poisson algebra associated with a finite-dimensional symplectic manifold ("phase space"). His algorithm gives a non-commutative, but associative, product (a so-called "star-product") between smooth phase space functions parameterized by Planck's constant $\hbar$, which is treated as a deformation parameter. In the limit as $\hbar$ goes to zero, the star product commutator goes to $\hbar$ times the Poisson bracket, so in this sense his method provides a quantization of the algebra of classical observables. In this work, we develop a generalization of Fedosov's method which applies to the infinite-dimensional symplectic "manifolds" that occur in Lagrangian field theories. We show that the procedure remains mathematically well-defined, and we explain the relationship of this method to more standard perturbative quantization schemes in quantum field theory.

Commutative Algebra I

Fedosov Quantization and Perturbative Quantum Field Theory

background

On maximal immediate extensions of valued fields: Maximal immediate extensions

We consider higher secant varieties to Veronese varieties. Most points on the r-th secant variety are represented by a finite scheme of length r contained in the Veronese variety --- in fact, for generic point, it is just a union of r distinct points. A modern way to phrase it is: the smoothable rank is equal to the border rank for most polynomials. This property is very useful for studying secant varieties, especially, whenever the smoothable rank is equal to the border rank for all points of the secant variety in question. In this note we investigate those special points for which the smoothable rank is not equal to the border rank. In particular, we show an explicit example of a cubic in five variables with border rank 5 and smoothable rank 6. We also prove that all cubics in at most four variables have the smoothable rank equal to the border rank.

On differences between the border rank and the smoothable rank of a polynomial

/pdf/on-elimination-of-variables-in-the-study-of-singularities-in-76ii3su97b.pdf

On elimination of variables in the study of singularities in positive characteristic

Under general conditions, the equation g(x, y) = 0 implicitly defines y locally as a function of x. In this article, we express divided differences of y in terms of bivariate divided differences of g, generalizing a recent result on divided differences of inverse functions.

https://www.duo.uio.no/bitstream/handle/10852/10725/dravhandling-muntingh.pdf?sequence=1

Topics in Polynomial Interpolation Theory

/pdf/commutative-algebra-i-47iyd2oh7q.pdf

* Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin Rings * Discrete Valuation Rings and Dedekind Domains * Completions * Dimension Theory

Commutative Algebra I

Citations

Fedosov Quantization and Perturbative Quantum Field Theory

On maximal immediate extensions of valued fields: Maximal immediate extensions

On differences between the border rank and the smoothable rank of a polynomial

On elimination of variables in the study of singularities in positive characteristic

Topics in Polynomial Interpolation Theory

References

Introduction to Commutative Algebra

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