Extensions of the Jacobi Identity for Vertex Operators, and Standard $A^{(1)}_1$-Modules
Extensions of the Jacobi Identity for Vertex Operators, and Standard $A^{(1)}_1$-Modules

Author: Cristiano Husu

Publisher: American Mathematical Soc.

Published: 1993

Total Pages: 98

ISBN-13: 0821825712

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The main axiom for a vertex operator algebra (over a field of characteristic zero), the Jacobi identity, is extended to multi-operator identities. Then relative [bold capital]Z2-twisted vertex operators are introduced and a Jacobi identity for these operators is established. Then these ideas are used to interpret and recover the twisted [bold capital]Z-operators and corresponding generating function identities developed by Lepowsky and R. L. Wilson. This work is closely related to the twisted parafermion algebra constructed by Zamolodchikov-Fateev.

$(16,6)$ Configurations and Geometry of Kummer Surfaces in ${\mathbb P}^3$
$(16,6)$ Configurations and Geometry of Kummer Surfaces in ${\mathbb P}^3$

Author: Maria del Rosario Gonzalez-Dorrego

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 114

ISBN-13: 0821825747

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The philosophy of the first part of this work is to understand (and classify) Kummer surfaces by studying (16, 6) configurations. Chapter 1 is devoted to classifying (16, 6) configurations and studying their manifold symmetries and the underlying questions about finite subgroups of [italic capitals]PGL4([italic]k). In chapter 2 we use this information to give a complete classification of Kummer surfaces together with explicit equations and the explicit description of their singularities.

Behavior of Distant Maximal Geodesics in Finitely Connected Complete 2-dimensional Riemannian Manifolds
Behavior of Distant Maximal Geodesics in Finitely Connected Complete 2-dimensional Riemannian Manifolds

Author: Takashi Shioya

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 90

ISBN-13: 082182578X

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This monograph studies the topological shapes of geodesics outside a large compact set in a finitely connected, complete, and noncompact surface admitting total curvature. When the surface is homeomorphic to a plane, all such geodesics behave like those of a flat cone. In particular, the rotation numbers of the geodesics are controlled by the total curvature. Accessible to beginners in differential geometry, but also of interest to specialists, this monograph features many illustrations that enhance understanding of the main ideas.

Elliptic Regularization and Partial Regularity for Motion by Mean Curvature
Elliptic Regularization and Partial Regularity for Motion by Mean Curvature

Author: Tom Ilmanen

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 106

ISBN-13: 0821825828

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We study Brakke's motion of varifolds by mean curvature in the special case that the initial surface is an integral cycle, giving a new existence proof by mean of elliptic regularization. Under a uniqueness hypothesis, we obtain a weakly continuous family of currents solving Brakke's motion. These currents remain within the corresponding level-set motion by mean curvature, as defined by Evans-Spruck and Chen-Giga-Goto. Now let [italic capital]T0 be the reduced boundary of a bounded set of finite perimeter in [italic capital]R[superscript italic]n. If the level-set motion of the support of [italic capital]T0 does not develop positive Lebesgue measure, then there corresponds a unique integral [italic]n-current [italic capital]T, [partial derivative/boundary/degree of a polynomial symbol][italic capital]T = [italic capital]T0, whose time-slices form a unit density Brakke motion. Using Brakke's regularity theorem, spt [italic capital]T is smooth [script capital]H[superscript italic]n-almost everywhere. In consequence, almost every level-set of the level-set flow is smooth [script capital]H[superscript italic]n-almost everywhere in space-time.

Some Special Properties of the Adjunction Theory for $3$-Folds in $\mathbb P^5$
Some Special Properties of the Adjunction Theory for $3$-Folds in $\mathbb P^5$

Author: Mauro Beltrametti

Publisher: American Mathematical Soc.

Published: 1995

Total Pages: 79

ISBN-13: 0821802348

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This work studies the adjunction theory of smooth 3-folds in P]5. Because of the many special restrictions on such 3-folds, the structure of the adjunction theoretic reductions are especially simple, e.g. the 3-fold equals its first reduction, the second reduction is smooth except possibly for a few explicit low degrees, and the formulae relating the projective invariants of the given 3-fold with the invariants of its second reduction are very explicit. Tables summarizing the classification of such 3-folds up to degree 12 are included. Many of the general results are shown to hold for smooth projective n-folds embedded in P]N with N 2n -1.

Iterating the Cobar Construction
Iterating the Cobar Construction

Author: Justin R. Smith

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 154

ISBN-13: 0821825887

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This paper develops a new invariant of a CW-complex called the m-structure and uses it to perform homotopy-theoretic computations. The m-structure of a space encapsulates the coproduct structure, as well as higher-coproduct structures that determine Steenrod-operations. Given an m-structure on the chain complex of a reduced simplicial complex of a pointed simply-connected space, one can equip the cobar construction of this chain-complex with a natural m-structure. This result allows one to form iterated cobar constructions that are shown to be homotopy equivalent to iterated loop-spaces.