Topology in Condensed Matter

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.