We examine the general problem of computing characteristic invariants of principal bundles whose structural group [italic capital]G is a topological group. Under the hypothesis that [italic capital]G has real cohomology finitely generated as an [bold]R-module, we are able to give a completely topological, local method for computing representative cocycles for real characteristic classes; our method applies, for example, to the (homologically) 10-dimensional non-Lie group of Hilton-Roitberg-Stasheff.

This work develops a topological analogue of the classical Chern-Weil theory as a method for computing the characteristic classes of principal bundles whose structural group is not necessarily a Lie group, but only a cohomologically finite topological group. Substitutes for the tools of differential geometry, such as the connection and curvature forms, are taken from algebraic topology, using work of Adams, Brown, Eilenberg-Moore, Milgram, Milnor and Stasheff. The result is a synthesis of the algebraic-topological and differential-geometric approaches to characteristic classes. In contrast to the first approach, specific cocycles are used, so as to highlight the influence of local geometry on global topology. In contrast to the second, calculations are carried out at the small scale rather than the infinitesimal; in fact, this work may be viewed as a systematic extension of the observation that curvature is the infinitesimal form of the defect in parallel translation around a rectangle. This book could be used as a text for an advanced graduate course in algebraic topology.

This monograph is based on lectures on topological quantum field theory given in 1989 at Princeton University by E. Witten, in which he unified several important mathematical works in terms of the Donaldson polynomial, Gromov/Floer homology, and Jones polynomials. Witten explained his three-dimensional construction of Jones polynomials, "an elegant construction of a new polynomial invariant in three-dimensional space" (per the author), via quantization of Chern-Simons gauge theory. Hu (Princeton U.) adds missing details and some new developments in the field. Annotation copyrighted by Book News Inc., Portland, OR.

The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a fundamental and an essential tool in the study of differentiable manifolds. In this volume, the authors provide a thorough introduction to characteristic classes, with detailed studies of Stiefel-Whitney classes, Chern classes, Pontrjagin classes, and the Euler class. Three appendices cover the basics of cohomology theory and the differential forms approach to characteristic classes, and provide an account of Bernoulli numbers. Based on lecture notes of John Milnor, which first appeared at Princeton University in 1957 and have been widely studied by graduate students of topology ever since, this published version has been completely revised and corrected.

This volume, recording the 10th international symposium honoring noted French mathematical physicist Jean-Pierre Vigier surveys and continues to develop Unified Field Mechanics (UFM) from the perspective of Multiverse cosmology and Topological Field Theory. UFM represents a developing paradigm shift with many new parameters extending the Standard Model to a 3rd regime of Natural Science beyond Quantum Mechanics. UFM is now experimentally testable, thus putatively able to demonstrate the existence of large-scale additional dimensionality (LSXD), test for QED violating phenomena and surmount the quantum uncertainty principle leading to a new 'Age of Discovery' paling all prior ages in the historical progression: Classical Mechanics (3D) to Quantum Mechanics (4D) and now to the birth of the 3rd regime of UFM in additional dimensionality correlating with M-Theory. Many still consider the Planck-scale as the 'basement of reality'. This could only be considered true under the limitations of the Standard Model. As we methodically enter the new regime a profound understanding of the multiverse and additional dimensionality beckons.

This is the third volume of the Handbook of Geometry and Topology of Singularities, a series which aims to provide an accessible account of the state of the art of the subject, its frontiers, and its interactions with other areas of research. This volume consists of ten chapters which provide an in-depth and reader-friendly survey of various important aspects of singularity theory. Some of these complement topics previously explored in volumes I and II, such as, for instance, Zariski’s equisingularity, the interplay between isolated complex surface singularities and 3-manifold theory, stratified Morse theory, constructible sheaves, the topology of the non-critical levels of holomorphic functions, and intersection cohomology. Other chapters bring in new subjects, such as the Thom–Mather theory for maps, characteristic classes for singular varieties, mixed Hodge structures, residues in complex analytic varieties, nearby and vanishing cycles, and more. Singularities are ubiquitous in mathematics and science in general. Singularity theory interacts energetically with the rest of mathematics, acting as a crucible where different types of mathematical problems interact, surprising connections are born and simple questions lead to ideas which resonate in other parts of the subject, and in other subjects. Authored by world experts, the various contributions deal with both classical material and modern developments, covering a wide range of topics which are linked to each other in fundamental ways. The book is addressed to graduate students and newcomers to the theory, as well as to specialists who can use it as a guidebook.

This book provides an accessible introduction to algebraic topology, a field at the intersection of topology, geometry and algebra, together with its applications. Moreover, it covers several related topics that are in fact important in the overall scheme of algebraic topology. Comprising eighteen chapters and two appendices, the book integrates various concepts of algebraic topology, supported by examples, exercises, applications and historical notes. Primarily intended as a textbook, the book offers a valuable resource for undergraduate, postgraduate and advanced mathematics students alike. Focusing more on the geometric than on algebraic aspects of the subject, as well as its natural development, the book conveys the basic language of modern algebraic topology by exploring homotopy, homology and cohomology theories, and examines a variety of spaces: spheres, projective spaces, classical groups and their quotient spaces, function spaces, polyhedra, topological groups, Lie groups and cell complexes, etc. The book studies a variety of maps, which are continuous functions between spaces. It also reveals the importance of algebraic topology in contemporary mathematics, theoretical physics, computer science, chemistry, economics, and the biological and medical sciences, and encourages students to engage in further study.

The volume is a collection of refereed research papers on infinite dimensional groups and manifolds in mathematics and quantum physics. Topics covered are: new classes of Lie groups of mappings, the Burgers equation, the Chern--Weil construction in infinite dimensions, the hamiltonian approach to quantum field theory, and different aspects of large N limits ranging from approximation methods in quantum mechanics to modular forms and string/gauge theory duality. Directed at research mathematicians and theoretical physicists as well as graduate students, the volume gives an overview of important themes of research at the forefront of mathematics and theoretical physics.

Over the past twenty-five years, mathematical concepts associated with geometric phases have come to occupy a central place in our modern understanding of the physics of electrons in solids. These 'Berry phases' describe the global phase acquired by a quantum state as the Hamiltonian is changed. Beginning at an elementary level, this book provides a pedagogical introduction to the important role of Berry phases and curvatures, and outlines their great influence upon many key properties of electrons in solids, including electric polarization, anomalous Hall conductivity, and the nature of the topological insulating state. It focuses on drawing connections between physical concepts and provides a solid framework for their integration, enabling researchers and students to explore and develop links to related fields. Computational examples and exercises throughout provide an added dimension to the book, giving readers the opportunity to explore the central concepts in a practical and engaging way.

There are three chapters to the memoir. The first defines and develops the notion of the index of a graph. The next chapter presents the general application of the graph index to knot theory. The last section is devoted to particular examples, such as determining the braid index of alternating pretzel links. A second result shows that for an alternating knot with Alexander polynomial having leading coefficient less than 4 in absolute value, the braid index is determined by polynomial invariants.