Principal Investigator: Jozef H. Przytycki, Mikhail G. Khovanov
This proposal asks for financial support for US based participants, for a 3-week long summer program devoted to Knot Theory: Knots in Poland III: conference on Knot Theory and its Ramifications. It will be a conference interspersed with specialized workshops and tutorials. It will be third conference in the series of large, very successful international conferences ``Knots in Poland". The conferences will take place in Warsaw July 18-25 and in Banach Center in Bedlewo July 25-August 4, 2010. Specifically, we propose to support about 15 graduate students and 10 mathematicians without grant. As part of the conference we will have a series of 3 talks by Tom Mrowka on his proof that Khovanov homology can be understood as a page in a spectral sequence converging to a version of Instanton Floer homology. A consequence of this is that Khovanov homology detects the unknot. Knots have fascinated people from the dawn of the human history. Much of the early knot theory was motivated by physics and chemistry (e.g. Kelvin theory of vortex atoms). The fundamental problem in knot theory is to be able to distinguish non-equivalent knots. There have been exciting new developments in the area of Knot Theory in recent years. From the Jones, Homflypt, and Kauffman polynomials, through quantum invariants of 3-manifolds, Topological Quantum Field Theories, to relations with gauge theory type invariants in 4-dimensional topology (Donaldson, Witten, etc). More recently, Khovanov introduced homology theory of links which categorifies the Jones polynomial (a chain complex is build on the Kauffman model of the Jones polynomial). Soon after, 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. One can mention here Khovanov-Rozansky categorification of Homflypt polynomial, categorification of the Kauffman bracket skein module of some 3-dimensional manifolds, and the result of T.Mrowka that Khovanov homology detects the unknot.
The subject has significant applications and relations with biology, physics, chemistry and the theory of computation. We feel that a conference on this subject is highly appropriate at this time, and we strive to be in the frontier of new development in knot theory and its ramifications. The project has broader impact in several aspects. Knots in Poland conference brings together from all over the word, third-world countries, former Soviet Union, minorities, women, 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. Distinguished researchers (including about 10 woman) will deliver plenary talks surveying the state of knowledge related to Knot Theory and its Ramifications. PhD students and fresh PhD's will be encouraged to attend. We expect to publish conference proceedings (as we did in the past) containing cutting-edge research papers and lecture notes which will be suitable for research mathematicians, students and readers with background in other exact sciences, including biology, chemistry, computer science, and physics.
Conference Web Page: www.mimuw.edu.pl/~traczyk/knotpol2010/<www.mimuw.edu.pl/%7Etraczyk/knotpol2010/>
Knot theory has long been an active branch of topology. There have been exciting new developments in the area of knot theory and 3-manifold topology in the last 27 years, from the Jones polynomial and 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. T.Mrowka and P.Kronheimer proved that Khovanov homology is an unknot detector. The new developments relate knot theory to remote branches of mathematics such as number theory, Lie theory, statistical physics, etc, and employ tools far beyond the traditional ones from algebraic topology. 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. Knots in Poland and Knots in Washington conferences contribute to the research in topology and knot theory in several ways. First of all, they create an environment for people to come together to exchange their latest results and produce new ideas. A few recent papers by several authors (e.g. Scott Carter, Stephan Wehrli, Hao Wu) give credit to our conferences as a location where the discussions that influenced their research took place. Our conferences provide opportunities for young people, including recent PhDs, graduate and even undergraduate students to be exposed to the cutting edge results in the current research frontier. This is crucial for the future of our discipline. Numerous talks at Knots in Poland and Knots in Washington conferences concern application of topology to other disciplines. Premiere example is the opening talk of Knots in Washington XXXII conference by Robert Ghrist: Towards Categorification in Applied Mathematics, or the talk by Robert Owczarek (Is quantum computing with superfluid helium possible: a potential role of Khovanov homology). One should also mention the talk by Karin Valencia (Topological characterization of knots and links arising as products of site-specific recombination on twist knots) at Knots in Poland. Several talks at Knots in Washington XXXII were devoted to Quantum Computing. In particular, plenary talks by Samson Abramsky and Bob Coecke were devoted to connections between Quantum Computing and Knot Theory. We have invited recognized leaders in the field, such as Anna Beliakova, Jim Cannon, Oliver Dasbach, Michael Eisermann, William Menasco, Hugh R. Morton, Tomasz Mrowka, Kunio Murasugi or Michael Polyak to give introductory research talks at Knots in Poland conference that serve as valuable educational resources for the attending students. Through this NSF award, we have provided support for many students (both graduate and undergraduate) to attend our conferences. It has been a valuable experience for them to listen to a variety of talks, talk to people working on various aspects of mathematics, present their own work, and to receive feedback from the experts. As one participating student wrote to us after the conference: `` I just wanted to say thanks for your tireless effort in running the Knots in Poland conference. I had a great time exploring Warsaw and Bedlewo, and I learned a lot of new information I'll be able to apply to my research. Giving my first conference talk was a good experience, and I was happy to be able to do so in front of a receptive international audience." Two volumes of Conference Proceedings will be published in Banach Center Publications. The first volume will appear in early 2012.
