ICTP Summer School and Conference Knot Theory; Spring 2009, Trieste, IL

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

Knot theory is very special topological subject: the classification of embeddings of a circle or collection of circles into three-dimensional space. This is a classical topological problem and a special case of the general placement problem: Understand the embeddings of a space X in another space Y. There have been exciting new developments in the area of knot theory and 3-manifold topology in the last 25 years. From the Jones, Homflypt and Kauffman polynomials, quantum invariants of 3-manifolds, through, Vassiliev invariants, topological quantum field theories, to relations with gauge theory type invariants in 4-dimensional topology (e.g. Donaldson, Witten). More recently, Khovanov introduced link homology as a generalization of the Jones polynomial to homology of chain complexes and Ozsvath and Szabo developed Heegaard-Floer homology, that lifts the Alexander polynomial. These two significantly different theories are closely related and the dependencies are the object of intensive study. These ideas mark the beginning of a new era in knot theory that includes relationships with four-dimensional problems and the creation of new forms of algebraic topology relevant to knot theory. It is a remarkable fact that knot theory, while very special in its problematic form, involves ideas and techniques that involve and inform much of mathematics and theoretical physics. The subject has significant applications and relations with biology, physics, combinatorics, algebra and the theory of computation. We feel that a summer school on this subject is highly appropriate at this time, and we strive to be in the frontier of new developments in knot theory and its applications. The project has several aspects. It is specialized in its concentration on the theory of knots. Knot theory, while a very focused mathematical activity has wide ramification in other sciences, particularly in physics and in biology and chemistry. Knots themselves occur physically in weaving and in the use of ropes from ancient times to the present day. In recent times, it has been suggested that knotted structures occur in quantum fields at the nuclear level, and knots have occurred significantly in quantum gravity theories and in string theory. In molecular biology DNA molecules can become knotted, and enzymes to avoid knotting are crucial for the replication of DNA and hence for the maintenance of life itself. In chemistry, molecules and long chain polymers can be knotted. The purpose of this summer school and conference is to examine both the applications and the theory of knots at this time. An ICTP Summer school and conference brings together participants from all over the world, third-world countries, women, minorities, and provides an opportunity for researchers and students to share their latest ideas and to collaborate with each other. The knowledge obtained on this occasion will be disseminated by participants throughout the world. All intensive short courses will be taught by experts, and lecture notes will be available online. The presentations include plenary talks by distinguished researchers, including at least 8 women, as well as short talks by other participants. We expect to publish conference proceedings containing cutting-edge research papers and lecture notes which will be suitable for research mathematicians, students, and readers in with background in other exact sciences, including biology, chemistry, computer science and physics.