Computational Geometry, Topology And Physics of

The Banff NATO Summer School was held August 14-25, 1989 at the Banff Cen­ tre, Banff, Albert, Canada. It was a combination of two venues: a summer school in the annual series of Summer School in Theoretical Physics spon­ sored by the Theoretical Physics Division, Canadian Association of Physi­ cists, and a NATO Advanced Study Institute. The Organizing Committee for the present school was composed of G. Kunstatter (University of Winnipeg), H.C. Lee (Chalk River Laboratories and University of Western Ontario), R. Kobes (University of Winnipeg), D.l. Toms (University of Newcastle Upon Tyne) and Y.S. Wu (University of Utah). Thanks to the group of lecturers (see Contents) and the timeliness of the courses given, the school, entitled PHYSICS, GEOMETRY AND TOPOLOGY, was popular from the very outset.

This book provides a complete introduction into spatial networks. It offers the mathematical tools needed to characterize these structures and how they evolve in time and presents the most important models of spatial networks.

Cyclic homology was introduced in the early eighties independently by Connes and Tsygan. They came from different directions. Connes wanted to associate homological invariants to K-homology classes and to describe the index pair­ ing with K-theory in that way, while Tsygan was motivated by algebraic K-theory and Lie algebra cohomology. At the same time Karoubi had done work on characteristic classes that led him to study related structures, without however arriving at cyclic homology properly speaking. Many of the principal properties of cyclic homology were already developed in the fundamental article of Connes and in the long paper by Feigin-Tsygan.

Two top experts in topology, O.Ya. Viro and D.B. Fuchs, give an up-to-date account of research in central areas of topology and the theory of Lie groups. They cover homotopy, homology and cohomology as well as the theory of manifolds, Lie groups, Grassmanians and low-dimensional manifolds. Their book will be used by graduate students and researchers in mathematics and mathematical physics.

This book provides a complete introduction into spatial networks. It offers the mathematical tools needed to characterize these structures and how they evolve in time and presents the most important models of spatial networks.

The general principles by which the editors and authors of the present edition have been guided were explained in the preface to the first volume of Mathemat­ ics of the 19th Century, which contains chapters on the history of mathematical logic, algebra, number theory, and probability theory (Nauka, Moscow 1978; En­ glish translation by Birkhiiuser Verlag, Basel-Boston-Berlin 1992). Circumstances beyond the control of the editors necessitated certain changes in the sequence of historical exposition of individual disciplines.

This book constitutes the proceedings of the 5th International Conference on Geometric Science of Information, GSI 2021, held in Paris, France, in July 2021.