Knots in Washington: Conferences on Knot Theory and its Ramifications

奖项：dms -1137422首席研究员：Jozef H. Przytycki, Valentina S. Harizanov, Alexander N. Shumakovitch， Hao Wu该奖项为一系列致力于结理论及其分支的会议的参与者提供部分支持。这是自1995年秋季以来在大华盛顿特区每学期举办的非常成功的“华盛顿结”系列会议的延续。这项建议要求在今后三年内支助四次会议。第一次这样的会议预计将在2011年秋季举行，重点是Khovanov同调，多架同调（如量子或布尔代数）和量子计算之间的联系，以及与分类的一般思想有关，包括串模的分类。这将是华盛顿会议系列的第33次会议。在过去的30年里，结理论和三流形拓扑领域有了令人兴奋的新发展。从3-流形的Jones多项式和量子不变量，通过Vassiliev不变量，拓扑量子场论，到四维拓扑中与规范论类型不变量的关系（如Donaldson， Witten）。最近，量子拓扑学又有了新的发展，以“Khovanov同调”的形式出现，将琼斯型多项式推广到链络合物的同调，显著地推广了不变量。此外，还提出了3-流形的Kauffman托架串模的分类，并对一些流形进行了发展。发现并研究了代数与Hochschild同调的关系。分析了计算Jones多项式和Turaev-Viro不变量的（量子）复杂性。“华盛顿结”会议将为上述领域的研究人员提供一个相互交流和合作的场所。杰出的研究人员将进行全体会谈，调查与霍万诺夫同调有关的知识状况。博士研究生和新博士将被鼓励参加。我们力求始终处于结理论及其分支的新发展的前沿。我们对会议主题的选择也反映了我们在乔治华盛顿大学的拓扑小组的实力。华盛顿的Knots为该领域的研究人员提供了一个分享最新想法和相互合作的机会。演讲包括杰出研究人员的全体会议，以及不同参与者的简短演讲。多年来，我们吸引了大量的博士后、研究生和一些本科生。在大华盛顿特区，华盛顿的Knots帮助汇集了相关的研究人员，并促进了该地区的研究活动。在国际上，华盛顿节吸引了来自世界各地的研究人员，现在被认为是结理论和低维拓扑的主要会议。我们期望出版会议记录（就像我们过去所做的那样），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://home.gwu.edu/~przytyck/knots/index.html

AbstractAward: DMS-1137422Principal Investigator: Jozef H. Przytycki, Valentina S. Harizanov, Alexander N. Shumakovitch, Hao Wu This award provides partial support for participants of a series of conferences devoted to knot theory and its ramifications. It is a continuation of the very successful series of the ``Knots in Washington'' conferences held every semester in the greater Washington, DC area since the Fall of 1995. This proposal asks for support for four conferences over the next three years. The first such conference is expected to be held in Fall of 2011, focusing on connections between Khovanov homology, homology of multi-shelves (e.g. quandles or Boolean algebras), and Quantum Computing, and relating to the general idea of categorification including categorification of skein modules. It will be the 33rd conference in the Knots in Washington Conference Series. There have been exciting new developments in the area of knot theory and 3-manifold topology in the last 30 years. From the Jones polynomial and 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, again, a new development in quantum topology has emerged in the form of ``Khovanov homology", a generalization of the Jones type polynomials to homology of chain complexes that significantly generalizes the invariants. Also, the categorification of the Kauffman bracket skein module of 3-manifolds has been proposed and developed for some manifolds. The relation to Hochschild homology of algebras has been discovered and studied. The (quantum) complexity of computing the Jones polynomial and Turaev-Viro invariants have been analyzed. "Knots in Washington" meetings would provide a place for researchers in described above areas to communicate and work with one another. Distinguished researchers will give plenary talks surveying the state of knowledge related to Khovanov homology. PhD students and fresh PhD's will be encouraged to attend. We strive to be always in the frontier of new developments in Knot Theory and its ramifications. Our choice of conference topics also reflects the strength of our topology group at the George Washington University.Knots in Washington provides an opportunity for researchers in the area to share their latest ideas and to collaborate with each other. The presentations include plenary talks by distinguished researchers, as well as short talks by various participants. Over the years, we have attracted a large number of postdoctors, graduate students, and some undergraduates. In the greater Washington DC area, Knots in Washington helps to bring together the relevant researchers, and to boost the research activities in the region. Internationally, Knots in Washington has attracted researchers from all over the world, and is now considered as a major conference in knot theory and low-dimensional topology. 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: http://home.gwu.edu/~przytyck/knots/index.html