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This lecture series was presented by a consortium of universities in conjunction with the U.S. Air Force Office of Scientific Research during the period 1967-1969 in Washington, D.C. and at the University of Maryland. The series of lectures was devoted to active basic areas of contemporary analysis which is important in or shows potential in real-world applications. Each lecture presents a survey and critical review of aspects of the specific area addressed, with emphasis on new results, open problems, and applications. This volume contains six lectures in the series; subsequent lectures will also be published.
The purpose of this book is to give background for those who would like to delve into some higher category theory. It is not a primer on higher category theory itself. It begins with a paper by John Baez and Michael Shulman which explores informally, by analogy and direct connection, how cohomology and other tools of algebraic topology are seen through the eyes of n-category theory. The idea is to give some of the motivations behind this subject. There are then two survey articles, by Julie Bergner and Simona Paoli, about (infinity,1) categories and about the algebraic modelling of homotopy n-types. These are areas that are particularly well understood, and where a fully integrated theory exists. The main focus of the book is on the richness to be found in the theory of bicategories, which gives the essential starting point towards the understanding of higher categorical structures. An article by Stephen Lack gives a thorough, but informal, guide to this theory. A paper by Larry Breen on the theory of gerbes shows how such categorical structures appear in differential geometry. This book is dedicated to Max Kelly, the founder of the Australian school of category theory, and an historical paper by Ross Street describes its development.
The Handbook of Homotopy Theory provides a panoramic view of an active area in mathematics that is currently seeing dramatic solutions to long-standing open problems, and is proving itself of increasing importance across many other mathematical disciplines. The origins of the subject date back to work of Henri Poincaré and Heinz Hopf in the early 20th century, but it has seen enormous progress in the 21st century. A highlight of this volume is an introduction to and diverse applications of the newly established foundational theory of ¥ -categories. The coverage is vast, ranging from axiomatic to applied, from foundational to computational, and includes surveys of applications both geometric and algebraic. The contributors are among the most active and creative researchers in the field. The 22 chapters by 31 contributors are designed to address novices, as well as established mathematicians, interested in learning the state of the art in this field, whose methods are of increasing importance in many other areas.
This book constitutes the thoroughly revised selected papers from the 17th International Symposium, FACS 2021, which was hel virtually in October 2021. The 7 full papers and 1 short contribution were carefully reviewed and selected from 16 submissions and are presented in the volume together with 1 invited paper. FACS 2021 is concerned with how formal methods can be applied to component-based software and system development. The book is subdivided into two blocks: Modelling & Composition and Verification. Chapter "A Linear Parallel Algorithm to Compute Bisimulation and Relational Coarsest Partitions" is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.