# New AMS-Distributed Publications

Number Theory

Computational Geometry of Positive Definite Quadratic Forms

Polyhedral Reduction Theories, Algorithms, and Applications

New AMS-Distributed Publications

Algebra and Algebraic Geometry

Achill Schürmann, Otto-von- Guericke Universität Magdeburg, Germany

Starting from classical arithmetical questions on quadratic forms, this book takes the reader step by step through the connections with lattice sphere packing and covering problems. As a model for polyhedral reduction theories of positive definite quadratic forms, Minkowski’s classical theory is presented, including an application to multidimensional continued fraction expansions. The reduction theories of Voronoi are described in great detail, including full proofs, new views, and generalizations that cannot be found elsewhere. Based on Voronoi’s second reduction theory, the local analysis of sphere coverings and several of its applications are presented. These include the classification of totally real thin number fields, connections to the Minkowski conjecture, and the discovery of new, sometimes surprising, properties of exceptional structures such as the Leech lattice or the root lattices.

Throughout this book, special attention is paid to algorithms and computability, allowing computer-assisted treatments. Although dealing with relatively classical topics that have been worked on extensively by numerous authors, this book is exemplary in showing how computers may help to gain new insights.

This item will also be of interest to those working in geometry and topology, algebra and algebraic geometry, and applications.

Contents: From quadratic forms to sphere packings and coverings; Minkowski reduction; Voronoi I; Voronoi II; Local analysis of coverings and applications; Polyhedral representation conversion under symmetries; Possible future projects; Bibliography; Index; Notations.

K-Theory and Noncommutative Geometry

Guillermo Cortiñas, University of Buenos Aires, Argentina, Joachim Cuntz, University of Münster, Munster, Germany, Max Karoubi, Université Paris VII, France, Ryszard Nest, University of Copenhagen, Denmark, and Charles A. Weibel, Rutgers University, New Brunswick, NJ, Editors

Since its inception 50 years ago, K-theory has been a tool for understanding a wide-ranging family of mathematical structures and their invariants: topological spaces, rings, algebraic varieties and operator algebras are the dominant examples. The invariants range from characteristic classes in cohomology, determinants of matrices, Chow groups of varieties, as well as traces and indices of elliptic operators. Thus K-theory is notable for its connections with other branches of mathematics.

Noncommutative geometry develops tools which allow one to think of noncommutative algebras in the same footing as commutative ones: as algebras of functions on (noncommutative) spaces. The algebras in question come from problems in various areas of mathematics and mathematical physics; typical examples include algebras of pseudodi erential operators, group algebras, and other algebras arising from quantum field theory.

University Lecture Series, Volume 48

January 2009, approximately 162 pages, Softcover, ISBN: 978-0- 8218-4735-0, LC 2008042435, 2000 Mathematics Subject Classifi- cation: 11-02, 52-02, 11Hxx, 52Bxx, 52Cxx, 90Cxx, 20H05; 11J70, 11R80, 20B25, 20H05, AMS members US$31, List US$39, Order code ULECT/48

To study noncommutative geometric problems one considers invariants of the relevant noncommutative algebras. These invariants include algebraic and topological K-theory, and also cyclic homology, discovered independently by Alain Connes and Boris Tsygan, which can be regarded both as a noncommutative version of de Rham cohomology and as an additive version of K-theory. There are primary and secondary Chern characters which pass from K-theory to cyclic homology. These characters are relevant both to noncommutative and commutative problems and have applications ranging from index theorems to the detection of singularities of commutative algebraic varieties.

The contributions to this volume represent this range of connections between K-theory, noncommmutative geometry, and other branches of mathematics.

This item will also be of interest to those working in analysis.

# January 2009

Notices of the AMS

79