Microlocal Analysis and Complex Fourier Analysis

This volume provides an introduction to knot and link invariants as generalized amplitudes for a quasi-physical process. The demands of knot theory, coupled with a quantum-statistical framework, create a context that naturally includes a range of interrelated topics in topology and mathematical physics. The author takes a primarily combinatorial stance toward knot theory and its relations with these subjects. This stance has the advantage of providing direct access to the algebra and to the combinatorial topology, as well as physical ideas. The book is divided into two parts: Part 1 is a systematic course on knots and physics starting from the ground up; and Part 2 is a set of lectures on various topics related to Part 1. Part 2 includes topics such as frictional properties of knots, relations with combinatorics and knots in dynamical systems. In this third edition, a paper by the author entitled "Knot Theory and Functional Integration" has been added. This paper shows how the Kontsevich integral approach to the Vassiliev invariants is directly related to the perturbative expansion of Witten's functional integral. While the book supplies the background, this paper can be read independently as an introduction to quantum field theory and knot invariants and their relation to quantum gravity. As in the second edition, there is a selection of papers by the author at the end of the book. Numerous clarifying remarks have been added to the text Vanishing of Stokes Curves (T Aoki et al.); Parabolic Equations with Singularity on the Boundary (C P Arceo et al.); Residues: Analysis or Algebra? (C A Berenstein); Heat Equation via Generalized Functions (S-Y Chung); Bergman Transformation for Analytic Functionals on Some Balls (K Fujita); On Infra-Red Singularities Associated with QC Photons (T Kawai & H P Stapp); Hyperfunctions and Kernel Method (D Kim); The Effect of New Stokes Curves in the Exact Steepest Descent Method (T Koike & Y Takei); Boehmians on the Sphere and Zonal Spherical Functions (M Morimoto); On a Generalization of the Laurent Expansion (Y Saburi); Domains of Convergence of Laplace Series (J Siciak); The Reproducing Kernels of the Space of Harmonic Polynomials in the Case of Real Rank 1 (R Wada & Y Agaoka); and other papers