Knots in Poland III; the conference on Knot Theory and its Ramifications

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AbstractAward: DMS-1034753Principal Investigator: Jozef H. Przytycki, Mikhail G. KhovanovThis 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 withbiology, physics, chemistry and the theory of computation. Wefeel that a conference on this subject is highly appropriate atthis time, and we strive to be in the frontier of new developmentin knot theory and its ramifications. The project has broaderimpact in several aspects. Knots in Poland conference bringstogether from all over the word, third-world countries, formerSoviet Union, minorities, women, and provides an opportunity forresearchers and students to share their latest ideas and tocollaborate with each other. The knowledge obtained on thisoccasion will be disseminated by participants throughout theworld. Distinguished researchers (including about 10 woman) willdeliver plenary talks surveying the state of knowledge related toKnot Theory and its Ramifications. PhD students and fresh PhD'swill be encouraged to attend. We expect to publish conferenceproceedings (as we did in the past) containing cutting-edgeresearch papers and lecture notes which will be suitable forresearch mathematicians, students and readers with background inother exact sciences, including biology, chemistry, computerscience, and physics. Conference Web Page:http://www.mimuw.edu.pl/~traczyk/knotpol2010/http://www.mimuw.edu.pl/%7Etraczyk/knotpol2010/