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

摘要奖：DMS-1034753主要研究员：Jozef H.Przytycki，Mikhail G.Khovanov这项提案要求为美国的参与者提供财政支持，用于为期3周的夏季项目，致力于结理论：波兰的结III：结理论及其分支会议。这将是一个穿插着专门讲习班和教程的会议。这将是非常成功的一系列大型国际会议中的第三次会议“波兰之结”。会议将于2010年7月18日至25日在华沙和7月25日至8月4日在贝德勒沃的Banach中心举行。具体地说，我们建议资助大约15名研究生和10名数学家，而不提供助学金。作为会议的一部分，我们将由Tom Mrowka进行一系列3次演讲，讨论他证明Khovanov同调可以理解为收敛到Instanton Floer同调的一个版本的谱序列中的一页。这样做的一个结果是，霍瓦诺夫同调检测到了这个结。从人类历史的黎明开始，结就让人们着迷。早期的纽结理论很大程度上是由物理和化学(如开尔文涡旋原子理论)推动的。纽结理论中的基本问题是能够区分不等价的纽结。近年来，纽结理论领域有了令人振奋的新发展。从Jones、Homflypt和Kauffman多项式，到三维流形的量子不变量、拓扑量子场论，再到与四维拓扑中规范理论类型不变量的关系(Donaldson，Witten等)。最近，Khovanov引入了链环的同调理论，它对Jones多项式进行了分类(链复合体建立在Jones多项式的Kauffman模型上)。不久之后，Ozsvath和Szabo发展了Heegaard-Floer同调，提升了Alexander多项式。这两种截然不同的理论有着密切的联系和相互依存的关系，是深入研究的对象。这里可以提到Homflypt多项式的Khovanov-Rozansky分类，一些三维流形的Kauffman括号斜线模的分类，以及T.Mrowka的结果Khovanov同调检测纽结。本课题与生物学、物理学、化学和计算理论有重要的应用和联系。我们认为，在这个时候召开一次关于这个主题的会议是非常合适的，我们努力站在新发展的结理论及其分支的前沿。该项目在几个方面产生了更广泛的影响。波兰结会议汇集了来自世界各地、第三世界国家、前苏联、少数民族和妇女的人们，并为研究人员和学生提供了一个分享他们最新想法和相互合作的机会。在这次会议上获得的知识将由与会者在世界各地传播。杰出的研究人员(包括大约10名女性)将发表全体会议，调查与纽结理论及其分支有关的知识状况。将鼓励博士生和应届博士生参加。我们希望发表会议记录(就像我们过去做的那样)，其中包括尖端的研究论文和课堂讲稿，这些将适合研究数学家、学生和具有其他精确科学背景的读者，包括生物、化学、计算机科学和物理。会议网络Page:http://www.mimuw.edu.pl/~traczyk/knotpol2010/http://www.mimuw.edu.pl/%7Etraczyk/knotpol2010/

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/