Woods Hole Mathematics

The following sections are included:

• PREFACE

• INTRODUCTION

• CONTENTS

We construct the mapping class group transformations that satisfy the pentagon relation for classical and quantum Teichmüller spaces coordinatized in terms of graphs. We derive classical and quantum geodesic algebras governed by the corresponding skein relations.

The aim of the lectures was to provide sufficient background to discuss recent work, done with Richard Melrose and Gerardo Mendoza on index formulae for Fredholm operators in the Heisenberg calculus. We review the geometry of strictly pseudoconvex CR-manifolds, contact manifolds and Grauert tubes. The calculus of Kohn-Nirenberg pseudodifferential operators is briefly described as well as the basic features of the Heisenberg calculus. A new proof of the Boutet de Monvel index theorem for Toeplitz operators is explained, as well as index theorems for certain classes of elliptic Heisenberg operators, and certain special classes of Fourier Integral Operators. Complete proofs of the results described in these lectures and considerable extensions of these results are found in the monograph by Melrose and myself.¹ These are notes for lectures that I presented at the summer school held at CIRM in Luminy, June 29-July 9, 1999 and Woods Hole, MA October 1999. I have added additional material to make the notes more comprehensible and self contained as well as some additional results in the last two sections.

In this paper we explore the boundary between biology, topology, algebra and the study of formal systems (logic).

We review the relationship of moduli spaces of surfaces to operads and deformations of algebras. We start by recalling basic facts about operads, examples of them, and their relations to algebras. In particular, we regard the Arc operad as well as several suboperads which can be thought of as cacti operads. These play an essential role in the string topology of Chas and Sullivan. It is recalled that spineless cacti and cacti homotopy equivalent to the little discs and framed little discs operads featured prominently in deformation theory. Furthermore, we analyze operations on operads and use the results to relate the spineless cacti operad to the renormalization Hopf algebra of Connes and Kreimer. Finally, we give a cell decomposition for spineless cacti and show that the cellular chains operate on the Hochschild complex of an associative algebra. This gives a solution to Deligne's conjecture about the Hochschild complex and furthermore directly relates the Gerstenhaber structures on the loop space and the Hochschild complex.

After giving a short survey of the theory as it exists to-date, we discuss the specifity of what we call the nonlinear version of Grothendieck-Teichmüller theory, in contrast with the pronilpotent motivic approach.

This survey covers earlier work of the author as well as recent work on Riemann's moduli space, its canonical cell decomposition and compactification, and the related operadic structure of arc complexes.

Turbulence is characterized by its intermittency, which is defined classically as localized bursts of small-scale activity in the observed quantity (e.g. velocity, vorticity, dissipation). Consequently, the simultaneous space and scale localization of the wavelet representation makes it a natural choice for studying intermittency. We propose different measures of intermittency, based on orthogonal wavelets, which avoid some problems associated with classical measures. We then apply them to study the intermittency of two-dimensional turbulence computed by direct numerical simulation.

The following sections are included:

• Introduction

• References

The following sections are included:

• SERIES ON KNOTS AND EVERYTHING