The contributions in this volume—dedicated to the work and mathematical interests of Oleg Viro on the occasion of his 60th birthday—are invited papers from the Marcus Wallenberg symposium and focus on research topics that bridge the gap among analysis, geometry, and topology. The encounters among these three fields are widespread and often provide impetus for major breakthroughs in applications.

Topics covered include new developments in low-dimensional topology related to invariants of links and three- and four-dimensional manifolds and the recent advances made in algebraic, complex, symplectic, and tropical geometry.

This collection is intended for graduate students and researchers in analysis, geometry, and topology, especially those whose area of study applies to two or more of these fields.

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 mappingclasses 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.

The contributions in this volume—dedicated to the work and mathematical interests of Oleg Viro on the occasion of his 60th birthday—are invited papers from the Marcus Wallenberg symposium and focus on research topics that bridge the gap among analysis, geometry, and topology. The encounters among these three fields are widespread and often provide impetus for major breakthroughs in applications.

Topics covered include new developments in low-dimensional topology related to invariants of links and three- and four-dimensional manifolds and the recent advances made in algebraic, complex, symplectic, and tropical geometry.

This collection is intended for graduate students and researchers in analysis, geometry, and topology, especially those whose area of study applies to two or more of these fields.

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 mappingclasses 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.