Ten years have passed since the publication of the first English edition. "Ten years is a long time, even when you are not in jail. " If we recall that the article on Riemann appeared in (Nature) in 1976, this Russian vaudeville joke is more than - propriate. Only the author's youth can account for the insane enterprise of presenting the scientific achievements and the biography of Riemann in 60 pages and the c- nection between physics and topology in the same space. Judging from the fact that the book sold out and got favorable reviews, there is a demand for publications of this type. It seems to me that popularizations aimed not at a narrow specialist but at a broader reader are especially needed nowadays. Specialists obtain information in their fields almost instantly thanks to the Internet, but to find out what is happening in contiguous fields, what problems and results are of great interest here, is not at all easy. Returning to my own book, I note with a certain pride that at least I seem to have evaluated accurately the trends in the development of theoretical physics. It is the combination of physics with topology and algebraic geometry that has led to the brightest achievements of mathematical physics in the last decade. It suffices to note the remarkable results of S.

Ten years have passed since the publication of the first English edition. "Ten years is a long time, even when you are not in jail. " If we recall that the article on Riemann appeared in (Nature) in 1976, this Russian vaudeville joke is more than - propriate. Only the author''s youth can account for the insane enterprise of presenting the scientific achievements and the biography of Riemann in 60 pages and the c- nection between physics and topology in the same space. Judging from the fact that the book sold out and got favorable reviews, there is a demand for publications of this type. It seems to me that popularizations aimed not at a narrow specialist but at a broader reader are especially needed nowadays. Specialists obtain information in their fields almost instantly thanks to the Internet, but to find out what is happening in contiguous fields, what problems and results are of great interest here, is not at all easy. Returning to my own book, I note with a certain pride that at least I seem to have evaluated accurately the trends in the development of theoretical physics. It is the combination of physics with topology and algebraic geometry that has led to the brightest achievements of mathematical physics in the last decade. It suffices to note the remarkable results of S.

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.

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.

Ten years have passed since the publication of the first English edition. "Ten years is a long time, even when you are not in jail. " If we recall that the article on Riemann appeared in (Nature) in 1976, this Russian vaudeville joke is more than - propriate. Only the author's youth can account for the insane enterprise of presenting the scientific achievements and the biography of Riemann in 60 pages and the c- nection between physics and topology in the same space. Judging from the fact that the book sold out and got favorable reviews, there is a demand for publications of this type. It seems to me that popularizations aimed not at a narrow specialist but at a broader reader are especially needed nowadays. Specialists obtain information in their fields almost instantly thanks to the Internet, but to find out what is happening in contiguous fields, what problems and results are of great interest here, is not at all easy. Returning to my own book, I note with a certain pride that at least I seem to have evaluated accurately the trends in the development of theoretical physics. It is the combination of physics with topology and algebraic geometry that has led to the brightest achievements of mathematical physics in the last decade. It suffices to note the remarkable results of S.

Ten years have passed since the publication of the first English edition. "Ten years is a long time, even when you are not in jail. " If we recall that the article on Riemann appeared in (Nature) in 1976, this Russian vaudeville joke is more than - propriate. Only the author''s youth can account for the insane enterprise of presenting the scientific achievements and the biography of Riemann in 60 pages and the c- nection between physics and topology in the same space. Judging from the fact that the book sold out and got favorable reviews, there is a demand for publications of this type. It seems to me that popularizations aimed not at a narrow specialist but at a broader reader are especially needed nowadays. Specialists obtain information in their fields almost instantly thanks to the Internet, but to find out what is happening in contiguous fields, what problems and results are of great interest here, is not at all easy. Returning to my own book, I note with a certain pride that at least I seem to have evaluated accurately the trends in the development of theoretical physics. It is the combination of physics with topology and algebraic geometry that has led to the brightest achievements of mathematical physics in the last decade. It suffices to note the remarkable results of S.

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.

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.