This IMA Volume in Mathematics and its Applications TOWARDS HIGHER CATEGORIES contains expository and research papers based on a highly successful IMA Summer Program on n-Categories: Foundations and Applications. We are grateful to all the participants for making this occasion a very productive and stimulating one. We would like to thank John C. Baez (Department of Mathematics, University of California Riverside) and J. Peter May (Department of Ma- ematics, University of Chicago) for their superb role as summer program organizers and editors of this volume. We take this opportunity to thank the National Science Foundation for its support of the IMA. Series Editors Fadil Santosa, Director of the IMA Markus Keel, Deputy Director of the IMA v PREFACE DEDICATED TO MAX KELLY, JUNE 5 1930 TO JANUARY 26 2007. This is not a proceedings of the 2004 conference "n-Categories: Fo- dations and Applications" that we organized and ran at the IMA during the two weeks June 7-18, 2004! We thank all the participants for helping make that a vibrant and inspiring occasion. We also thank the IMA sta? for a magni?cent job. There has been a great deal of work in higher c- egory theory since then, but we still feel that it is not yet time to o?er a volume devoted to the main topic of the conference.

This is an introduction to the basic tools of mathematics needed to understand the relation between knot theory and quantum gravity. The book begins with a rapid course on manifolds and differential forms, emphasizing how these provide a proper language for formulating Maxwell's equations on arbitrary spacetimes. The authors then introduce vector bundles, connections and curvature in order to generalize Maxwell theory to the Yang-Mills equations. The relation of gauge theory to the newly discovered knot invariants such as the Jones polynomial is sketched. Riemannian geometry is then introduced in order to describe Einstein's equations of general relativity and show how an attempt to quantize gravity leads to interesting applications of knot theory.

This is an introduction to the basic tools of mathematics needed to understand the relation between knot theory and quantum gravity. The book begins with a rapid course on manifolds and differential forms, emphasizing how these provide a proper language for formulating Maxwell's equations on arbitrary spacetimes. The authors then introduce vector bundles, connections and curvature in order to generalize Maxwell theory to the Yang-Mills equations. The relation of gauge theory to the newly discovered knot invariants such as the Jones polynomial is sketched. Riemannian geometry is then introduced in order to describe Einstein's equations of general relativity and show how an attempt to quantize gravity leads to interesting applications of knot theory.

This IMA Volume in Mathematics and its Applications TOWARDS HIGHER CATEGORIES contains expository and research papers based on a highly successful IMA Summer Program on n-Categories: Foundations and Applications. We are grateful to all the participants for making this occasion a very productive and stimulating one. We would like to thank John C. Baez (Department of Mathematics, University of California Riverside) and J. Peter May (Department of Ma- ematics, University of Chicago) for their superb role as summer program organizers and editors of this volume. We take this opportunity to thank the National Science Foundation for its support of the IMA. Series Editors Fadil Santosa, Director of the IMA Markus Keel, Deputy Director of the IMA v PREFACE DEDICATED TO MAX KELLY, JUNE 5 1930 TO JANUARY 26 2007. This is not a proceedings of the 2004 conference "n-Categories: Fo- dations and Applications" that we organized and ran at the IMA during the two weeks June 7-18, 2004! We thank all the participants for helping make that a vibrant and inspiring occasion. We also thank the IMA sta? for a magni?cent job. There has been a great deal of work in higher c- egory theory since then, but we still feel that it is not yet time to o?er a volume devoted to the main topic of the conference.

In the mid-1980's, Abhay Ashtekar discovered a reformulation of general relativity in terms of what are now called the "new variables". This led to a rapid series of developments, and in 1990 Carlo Rovelli and Lee Smolin published a paper entitled "Loop representation of quantum general relativity", in which, among other things, it was shown that there is a space of solutions to the constraint equations of quantum gravity consisting of link invariants. The discovery of relations between quantum gravity and the theory of knots and links came as quite a surprise, since the topics had been regarded as quite remote from each other. This volume is the proceedings of a workshop held in 1993 at the University of California at Riverside. The purpose of the workshop was to bring together researchers in knot theory and quantum gravity and form more bridges between the two subjects. Most of the talks were given by researchers whose work has significance for both subjects. This volume contains expository papers as well as new results, and should serve as a guide for mathematicians and physicists seeking to understand this rapidly developing area of research.

The authors present a rigorous treatment of the first principles of the algebraic and analytic core of quantum field theory. Their aim is to correlate modern mathematical theory with the explanation of the observed process of particle production and of particle-wave duality that heuristic quantum field theory provides. Many topics are treated here in book form for the first time, from the origins of complex structures to the quantization of tachyons and domains of dependence for quantized wave equations. This work begins with a comprehensive analysis, in a universal format, of the structure and characterization of free fields, which is illustrated by applications to specific fields. Nonlinear local functions of both free fields (or Wick products) and interacting fields are established mathematically in a way that is consistent with the basic physical constraints and practice. Among other topics discussed are functional integration, Fourier transforms in Hilbert space, and implementability of canonical transformations. The authors address readers interested in fundamental mathematical physics and who have at least the training of an entering graduate student. A series of lexicons connects the mathematical development with the underlying physical motivation or interpretation. The examples and problems illustrate the theory and relate it to the scientific literature.

Originally published in 1992.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

The authors present a rigorous treatment of the first principles of the algebraic and analytic core of quantum field theory. Their aim is to correlate modern mathematical theory with the explanation of the observed process of particle production and of particle-wave duality that heuristic quantum field theory provides. Many topics are treated here in book form for the first time, from the origins of complex structures to the quantization of tachyons and domains of dependence for quantized wave equations. This work begins with a comprehensive analysis, in a universal format, of the structure and characterization of free fields, which is illustrated by applications to specific fields. Nonlinear local functions of both free fields (or Wick products) and interacting fields are established mathematically in a way that is consistent with the basic physical constraints and practice. Among other topics discussed are functional integration, Fourier transforms in Hilbert space, and implementability of canonical transformations. The authors address readers interested in fundamental mathematical physics and who have at least the training of an entering graduate student. A series of lexicons connects the mathematical development with the underlying physical motivation or interpretation. The examples and problems illustrate the theory and relate it to the scientific literature. Originally published in 1992. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

We introduce the theory of chemical reaction networks and their relation to stochastic Petri nets - important ways of modeling population biology and many other fields. We explain how techniques from quantum mechanics can be used to study these models. This relies on a profound and still mysterious analogy between quantum theory and probability theory, which we explore in detail. We also give a tour of key results concerning chemical reaction networks and Petri nets.