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

摘要：项目编号：dms -1034753首席研究员：Jozef H. Przytycki, Mikhail G. khovanov本提案要求为美国参与者提供为期三周的暑期项目的资金支持，该项目致力于“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研讨会和教程的会议。这将是“波兰节”系列大型、非常成功的国际会议中的第三次会议。会议将于2010年7月18日至25日在华沙举行，7月25日至8月4日在贝德勒沃的巴拿赫中心举行。具体来说，我们建议无偿资助15名左右的研究生和10名左右的数学家。作为会议的一部分我们将有Tom Mrowka的三场演讲他证明了Khovanov同调可以被理解为光谱序列中的一页收敛于Instanton flower同调的一个版本。这样做的一个结果是，Khovanov同源性检测到解结。从人类历史的黎明开始，结就让人们着迷。许多早期的结理论是由物理和化学（例如开尔文的涡原子理论）推动的。结理论的基本问题是如何区分非等效结。近年来，在结理论领域有了令人兴奋的新发展。从Jones， Homflypt和Kauffman多项式，通过3流形的量子不变量，拓扑量子场论，到与4维拓扑中规范论类型不变量的关系（Donaldson， Witten等）。最近，Khovanov引入了对Jones多项式进行分类的链接同源理论（一个链复合体是建立在Jones多项式的Kauffman模型上的）。不久之后，Ozsvath和Szabo发展了Heegaard-Floer同调，从而提升了Alexander多项式。这两种截然不同的理论密切相关，相互依存，是深入研究的对象。这里可以提到Homflypt多项式的Khovanov- rozansky分类，某些三维流形的Kauffman支架串模的分类，以及T.Mrowka关于Khovanov同调检测解结的结果。这门学科与生物学、物理学、化学和计算理论有着重要的应用和联系。我们认为在这个时候召开这个主题的会议是非常合适的，我们努力走在结理论及其分支新发展的前沿。该项目在几个方面产生了广泛的影响。波兰节会议汇集了来自世界各地，第三世界国家，前苏联，少数民族，妇女，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。在这次活动中获得的知识将由世界各地的参与者传播。杰出的研究人员（包括大约10名女性）将发表全体会议，调查与结理论及其分支相关的知识状况。鼓励博士生和刚毕业的博士生参加。我们期望出版会议论文集（就像我们过去所做的那样），其中包含前沿的研究论文和课堂讲稿，适合研究数学家、学生和具有其他精确科学背景的读者，包括生物学、化学、计算机科学和物理学。会议网页：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/