Gorenstein Quotient Singularities in Dimension Three

Gorenstein Quotient Singularities in Dimension Three

Author: Stephen Shing-Toung Yau

Publisher: American Mathematical Soc.

Published: 1993

Total Pages: 102

ISBN-13: 0821825674

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In chapter one we address the classification of finite subgroups of [italic capitals]SL([bold]3,[double-struck capital]C). This is followed by a general method to find invariant polynomials and their relations of finite subgroups of [italic capitals]GL([bold]3,[double-struck capital]C). Lastly, we recall some properties of quotient varieties and prove that [double-struck capital]C3/[italic capital]G has isolated singularities if and only if [italic capital]G is abelian and 1 is not an eigenvalue of g in [italic capital]G.


Finite Rational Matrix Groups

Finite Rational Matrix Groups

Author: Gabriele Nebe

Publisher: American Mathematical Soc.

Published: 1995

Total Pages: 158

ISBN-13: 0821803433

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The study of finite rational matrix groups reduces to the investigation of the maximal finite irreducible matrix groups and their natural lattices, which often turn out to have rather beautiful geometric and arithmetic properties. This book presents a full classification in dimensions up to 23 and with restrictions in dimensions and p +1 and p-1 for all prime numbers p. Nonmaximal finite groups might act on several types of lattices and therefore embed into more than one maximal finite group. This gives rise to a simplicial complex interrelating the maximal finite groups and measuring the complexity of the dimension. Group theory, integral representation theory, arithmetic theory of quadratic forms and algorithmic methods are used.


Filtrations on the Homology of Algebraic Varieties

Filtrations on the Homology of Algebraic Varieties

Author: Eric M. Friedlander

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 126

ISBN-13: 0821825917

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This work provides a detailed exposition of a classical topic from a very recent viewpoint. Friedlander and Mazur describe some foundational aspects of ``Lawson homology'' for complex projective algebraic varieties, a homology theory defined in terms of homotopy groups of spaces of algebraic cycles. Attention is paid to methods of group completing abelian topological monoids. The authors study properties of Chow varieties, especially in connection with algebraic correspondences relating algebraic varieties. Operations on Lawson homology are introduced and analysed. These operations lead to a filtration on the singular homology of algebraic varieties, which is identified in terms of correspondences and related to classical filtrations of Hodge and Grothendieck.


Triangular Algebras and Ideals of Nest Algebras

Triangular Algebras and Ideals of Nest Algebras

Author: John Lindsay Orr

Publisher: American Mathematical Soc.

Published: 1995

Total Pages: 65

ISBN-13: 0821804057

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Immersive environments such as virtual reality technology makes possible can respond to their audiences, so that each person's experience of the environment is unique. This volume brings together 11 essays along with artists' projects produced at the Banff Centre for the Arts in Canada to explore issues raised by the creation of virtual environments. The essays approach the social and cultural implications of cyberspace from the perspective of cultural studies, communications, art history, art criticism, English, and women's studies; while artists who created nine virtual worlds at the Banff Centre discuss what they have tried to accomplish in both theoretical and technical terms. With 64 illustrations, including 18 color plates. Annotation copyright by Book News, Inc., Portland, OR


Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distributions

Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distributions

Author: Wensheng Liu

Publisher: American Mathematical Soc.

Published: 1995

Total Pages: 121

ISBN-13: 0821804049

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A sub-Riemannian manifold ([italic capitals]M, E, G) consists of a finite-dimensional manifold [italic capital]M, a rank-two bracket generating distribution [italic capital]E on [italic capital]M, and a Riemannian metric [italic capital]G on [italic capital]E. All length-minimizing arcs on ([italic capitals]M, E, G) are either normal extremals or abnormal extremals. Normal extremals are locally optimal, i.e., every sufficiently short piece of such an extremal is a minimizer. The question whether every length-minimizer is a normal extremal was recently settled by R. G. Montgomery, who exhibited a counterexample. The present work proves that regular abnormal extremals are locally optimal, and, in the case that [italic capital]E satisfies a mild additional restriction, the abnormal minimizers are ubiquitous rather than exceptional. All the topics of this research report (historical notes, examples, abnormal extremals, Hamiltonians, nonholonomic distributions, sub-Riemannian distance, the relations between minimality and extremality, regular abnormal extremals, local optimality of regular abnormal extremals, etc.) are presented in a very clear and effective way.


The Full Set of Unitarizable Highest Weight Modules of Basic Classical Lie Superalgebras

The Full Set of Unitarizable Highest Weight Modules of Basic Classical Lie Superalgebras

Author: Hans Plesner Jakobsen

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 129

ISBN-13: 0821825933

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This work contains a complete description of the set of all unitarizable highest weight modules of classical Lie superalgebras. Unitarity is defined in the superalgebraic sense, and all the algebras are over the complex numbers. Part of the classification determines which real forms, defined by anti-linear anti-involutions, may occur. Although there have been many investigations for some special superalgebras, this appears to be the first systematic study of the problem.


The Index Theorem for Minimal Surfaces of Higher Genus

The Index Theorem for Minimal Surfaces of Higher Genus

Author: Friedrich Tomi

Publisher: American Mathematical Soc.

Published: 1995

Total Pages: 90

ISBN-13: 0821803522

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In this paper we formulate and prove an index theorem for minimal surfaces of higher topological type spanning one boundary contour. Our techniques carry over to surfaces with several boundary contours as well as to unoriented surfaces.


Littlewood-Paley Theory on Spaces of Homogeneous Type and the Classical Function Spaces

Littlewood-Paley Theory on Spaces of Homogeneous Type and the Classical Function Spaces

Author: Yongsheng Han

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 138

ISBN-13: 0821825925

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In this work, Han and Sawyer extend Littlewood-Paley theory, Besov spaces, and Triebel-Lizorkin spaces to the general setting of a space of homogeneous type. For this purpose, they establish a suitable analogue of the Calder 'on reproducing formula and use it to extend classical results on atomic decomposition, interpolation, and T1 and Tb theorems. Some new results in the classical setting are also obtained: atomic decompositions with vanishing b-moment, and Littlewood-Paley characterizations of Besov and Triebel-Lizorkin spaces with only half the usual smoothness and cancellation conditions on the approximate identity.


Higher Spinor Classes

Higher Spinor Classes

Author: J. F. Jardine

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 101

ISBN-13: 0821825909

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This work defines the higher spinor classes of an orthogonal representation of a Galois group. These classes are higher-degree analogues of the Fröhlich spinor class, which quantify the difference between the Stiefel-Whitney classes of an orthogonal representation and the Hasse-Witt classes of the associated form. Jardine establishes various basic properties, including vanishing in odd degrees and an induction formula for quadratic field extensions. The methods used include the homotopy theory of simplicial presheaves and the action of the Steenrod algebra on mod 2 étale cohomology.


(m)KdV Solitons on the Background of Quasi-Periodic Finite-Gap Solutions

(m)KdV Solitons on the Background of Quasi-Periodic Finite-Gap Solutions

Author: Fritz Gesztesy

Publisher: American Mathematical Soc.

Published: 1995

Total Pages: 102

ISBN-13: 0821804065

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In the introductory section, we review the formulation of the Korteweg-de Vries (KdV) equation and of the modified KdV (mKdV) equation as a compatibility condition for a Lax pair of linear operators. We then illustrate Miura's transformation, which maps solutions of the mKdV into solutions of the KdV. We then give a general overview of the concept of soliton solutions relative to general backgrounds, and of the single and double commutation methods. Finally, we present the main results of the article. To avoid the clutter of too many technical details, the paper is organized in four sections and five appendices.