Burglind Jöricke

This book studies the relation between conformal invariants and dynamical invariants and their applications, taking the reader on an excursion through a wide range of topics. The conformal invariants, called here the conformal modules of conjugacy classes of elements of the fundamental group, were proposed by Gromov in the case of the twice punctured complex plane. They provide obstructions to Gromov's Oka Principle. The invariants of the space of monic polynomials of degree n appeared earlier in relation to Hilbert's 13th Problem, and are called the conformal modules of conjugacy classes of braids. Interestingly, the conformal module of a conjugacy class of braids is inversely proportional to a popular dynamical invariant, the entropy, which was studied in connection with Thurston's celebrated theory of surface homeomorphisms. This result, proved here for the first time, is another instance of the numerous manifestations of the unity of mathematics, and it has applications. After prerequisites on Riemann surfaces, braids, mapping classes and elements of Teichmüller theory, a detailed introduction to the entropy of braids and mapping classes is given, with thorough, sometimes new proofs. Estimates are provided of Gromov's conformal invariants of the twice punctured complex plane and it is shown that the upper and lower bounds differ by universal multiplicative constants. These imply estimates of the entropy of any pure three-braid, and yield quantitative statements on the limitations of Gromov's Oka Principle in the sense of finiteness theorems, using conformal invariants which are related to elements of the fundamental group (not merely to conjugacy classes). Further applications of the concept of conformal module are discussed. Aimed at graduate students and researchers, the book proposes several research problems.

Braids, Conformal Module, Entropy, and Gromov's Oka Principle

Victor Havin

The present book is a collection of variations on a theme which can be summed up as follows: It is impossible for a non-zero function and its Fourier transform to be simultaneously very small. In other words, the approximate equalities x :::::: y and x :::::: fj cannot hold, at the same time and with a high degree of accuracy, unless the functions x and yare identical. Any information gained about x (in the form of a good approximation y) has to be paid for by a corresponding loss of control on x, and vice versa. Such is, roughly speaking, the import of the Uncertainty Principle (or UP for short) referred to in the title ofthis book. That principle has an unmistakable kinship with its namesake in physics - Heisenberg's famous Uncertainty Principle - and may indeed be regarded as providing one of mathematical interpretations for the latter. But we mention these links with Quantum Mechanics and other connections with physics and engineering only for their inspirational value, and hasten to reassure the reader that at no point in this book will he be led beyond the world of purely mathematical facts. Actually, the portion of this world charted in our book is sufficiently vast, even though we confine ourselves to trigonometric Fourier series and integrals (so that "The U. P. in Fourier Analysis" might be a slightly more appropriate title than the one we chose).

The Uncertainty Principle in Harmonic Analysis

Ilia Itenberg

Mikael Passare

The articles in this volume are invited papers from the Marcus Wallenberg symposium and focus on research topics that bridge the gap between analysis, geometry, and topology. The encounters between these three fields are widespread and often provide impetus for major breakthroughs in applications. Topics include new developments in low dimensional topology related to invariants of links and three and four manifolds; Perelman's spectacular proof of the Poincare conjecture; and the recent advances made in algebraic, complex, symplectic, and tropical geometry.

Burglind Jöricke

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