This volume is based on the talks and lectures given by the participants of the 3-month seminar program "Topology in Condensed Matter," which was held in the MPIPKS Dresden, 8 May-31 July 2002 under the scienti?c direction of Professors M. Kleman, S. Novikov, and myself. The aim of this program was to discuss recent applications of topology to several areas in condensed matter physics and related ?elds like biology. The last 30 years of the development of modern physics a?rmed two important ideas. The ?rst is the e?cient applications of topology in physics. One should mention applications in condensed matter, such as classi?cation of defects and textures in liquid crystals and super?uid liquids, the role of entangibility in polymer physics and DNA structures. The second tendency is also very prevalent. Some important discoveries in particle physics and condensed m- ter led to new and unpredictable questions in pure mathematics. We refer to the invention of monopoles and instantons in quantum ?eld theory, q- sicrystals ?uid membranes of high genus, fullerenes (C ,C , etc. ), and so on 60 90 in condensed matter. The number of such applications in the last years has increased substantially. The papers presented in this volume and the next one "Topology in - ology" re?ect the spectrum of topics discussed. Besides original papers, a mini-course in topology for physicists and biologists was organized. These lectures will be published in the second volume.

This significantly expanded second edition of Riemann, Topology, and Physics combines a fascinating account of the life and work of Bernhard Riemann with a lucid discussion of current interaction between topology and physics, The author, a distinguished mathematical physicist, takes into account his own research at the Riemann archives of GAttingen University and developments over the last decade that connect Riemann with numerous significant ideas and methods reflected throughout contemporary mathematics and physics. Special attention is paid in part one to results on the Riemanna Hilbert problem and, in part two, to discoveries in field theory and condensed matter such as the quantum Hall effect, quasicrystals, membranes with nontrivial topology, fake differential structures on 4-dimensional Euclidean space, new invariants of knots and more.

The contents of this book focus on the recent investigations in molecular bi- ogywhereapplicationsoftopologyseemtobeverystimulating. Thevolumeis based on the talks and lectures given by participants of the three-month p- gram"TopologyinCondensedMatter,"whichwasheldintheMaxPlanck- stitut fur Physik komplexer Systeme, Dresden, Germany, 8May-31July 2002, under the scienti?c direction of Professors M. Kl eman, S. Novikov and - self. The aim of this program was to discuss recent applications of topology to several areas in condensed matter physics and molecular biology. The ?rst volume "Topology in Condensed Matter" is concerned with m- ern applications of geometrical and topological techniques to such new and classic ?elds of physics like electron theory of metals, theory of nano-crystals, aperiodic and liquid crystals, quantum computation and so on. This volume is published simultaneously in "Springer Series in Solid-State Physics." The present volume gives an exposition of the role of topology in the theory of proteins and DNA. The last thirty years a?rmed very e?cient - plications of modern mathematics, especially topology, in physics. The union of mathematics and physics was very stimulating for both sides. On the other hand, the impact of mathematics in biology has been rather limited. H- ever here also some interesting results were obtained. In particular, there are applications of knot theory in the theory of circular closed DNA. The - cent discoveries in molecular biology indicate future successful applications of topology."

This volume is based on the talks and lectures given by the participants of the 3-month seminar program "Topology in Condensed Matter," which was held in the MPIPKS Dresden, 8 May-31 July 2002 under the scienti?c direction of Professors M. Kleman, S. Novikov, and myself. The aim of this program was to discuss recent applications of topology to several areas in condensed matter physics and related ?elds like biology. The last 30 years of the development of modern physics a?rmed two important ideas. The ?rst is the e?cient applications of topology in physics. One should mention applications in condensed matter, such as classi?cation of defects and textures in liquid crystals and super?uid liquids, the role of entangibility in polymer physics and DNA structures. The second tendency is also very prevalent. Some important discoveries in particle physics and condensed m- ter led to new and unpredictable questions in pure mathematics. We refer to the invention of monopoles and instantons in quantum ?eld theory, q- sicrystals ?uid membranes of high genus, fullerenes (C ,C , etc. ), and so on 60 90 in condensed matter. The number of such applications in the last years has increased substantially. The papers presented in this volume and the next one "Topology in - ology" re?ect the spectrum of topics discussed. Besides original papers, a mini-course in topology for physicists and biologists was organized. These lectures will be published in the second volume.