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.

M.Monastyrsky is a professor of mathematics and theoretical physics
at the Institute of Theoretical and Experimental Physics,Moscow,Russia,
graduated  the mathematical department of Moscow state university(1967),
author of numerous papers in diferent fields of mathematics and
theoretical physics including the monograph :Topology in Gauge Fields
and Condensed Matter (Plenum,1993 ) and two other books:Riemann,Topology
Physics,Birkhauser,Boston,1999 ,2nd ed. and Modern mathematics in the
light of Fields medals,AKPeters,1997.I gave invited plenary talks on
many conferences and was the visiting professor of many universities
and institutes including IAS(Princeton),Harvard,Princeton and Yale
Universities (USA),Cambridge and Oxford Universities (Great Britain),
IHES and Universite de Paris (France).

Topology in Biology: From DNA Mechanics to Enzymology.- Monte Carlo Simulation of DNA Topological Properties.- Dynamics of DNA Supercoiling.- From Tangle Fractions to DNA.- Linear Behavior of the Writhe Versus the Number of Crossings in Rational Knots and Links.- Combinatories and Topology of the ?-Sandwich and ?-Barrel Proteins.- The Structure of Collagen.- Euler Characteristic, Dehn-Sommerville Characteristics, and Their Applications.- Hopf Fibration and Its Applications.- Multi-Valued Functionals, One-Forms and Deformed de Rham Complex.- The Spectral Geometry of Riemann Surfaces.