ICTP Summer School and Conference Knot Theory; Spring 2009, Trieste, IL

ICTP暑期学校和会议结理论；

• 批准号: 0925541
• 负责人: Louis Kauffman
• 金额: $ 2.8万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-05-01 至 2011-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0925541&HistoricalAwards=false
• 关键词: ICTP; Summer; Conference; Knot; Theory

Knot theory is very special topologicalsubject: the classification of embeddings of a circle or collection of circles into three-dimensional space. This is a classical topological problem and a special case of the general placement problem: Understand the embeddings of a space X in another space Y. There have been exciting new developments in the area of knot theory and 3-manifold topology in the last 25 years.From the Jones, Homflypt and Kauffman polynomials, 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, 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. These two significantly different theories are closely related and the dependencies are the object of intensive study. 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. It is a remarkable fact that knot theory, while very special in its problematic form, involves ideas and techniques that involve and inform much of mathematics and theoretical physics. The subject has significant applications and relations with biology, physics, combinatorics, algebra and the theory of computation. We feel that a summer school on this subject is highly appropriate at this time, and we strive to be in the frontier of new developments in knot theory and its applications.The project has several aspects. It is specialized in its concentration on the theory of knots. Knot theory, while a very focused mathematical activity has wide ramification in other sciences, particularly in physics and in biology and chemistry. Knots themselves occur physically in weaving and in the use of ropes from ancient times to the present day. In recent times, it has been suggested that knotted structures occur in quantum fields at the nuclear level, and knots have occurred significantly in quantum gravity theories and in string theory. In molecular biology DNA molecules can become knotted, and enzymes to avoid knotting are crucial for the replication of DNA and hence for the maintenance of life itself. In chemistry, molecules and long chain polymers can be knotted. The purpose of this summer school and conference is to examine both the applications and the theory of knots at this time. An ICTP Summer school and conference brings together participants from all over the world, third-world countries, women, minorities, 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. All intensive short courses will be taught by experts, and lecture notes will be available online. The presentations include plenary talks by distinguished researchers, including at least 8 women, as well as short talks by other participants. We expect to publish conference proceedings containing cutting-edge research papers and lecture notes which will be suitable for research mathematicians, students, and readers in with background in other exact sciences, including biology, chemistry, computer science and physics.

纽结理论是一门非常特殊的拓扑学科：它是对一个圆或一组圆嵌入三维空间的分类。这是一个经典的拓扑问题，也是一般布局问题的一个特例：理解一个空间X在另一个空间Y中的嵌入。在过去的25年里，纽结理论和三维流形拓扑领域有了令人兴奋的新发展，从Jones、Homflypt和Kauffman多项式，三维流形的量子不变量，到Vassiliev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（例如唐纳森，维滕）。 最近，Khovanov引入了链接同源性作为琼斯多项式到链复合物同源性的推广，Ozsvath和Szabo开发了Heegaard-Floer同源性，提升了亚历山大多项式。 这两个显著不同的理论是密切相关的，依赖关系是深入研究的对象。这些想法标志着纽结理论一个新时代的开始，包括与四维问题的关系，以及与纽结理论相关的代数拓扑的新形式的创造。一个值得注意的事实是，纽结理论虽然在其问题形式上非常特殊，但它所涉及的思想和技术涉及并告知了许多数学和理论物理学。这门学科与生物学、物理学、组合学、代数学和计算理论有着重要的应用和关系。我们认为，在这个时候，这个主题的暑期学校是非常合适的，我们努力在纽结理论及其应用的新发展的前沿。它是专门在其浓度上的理论结。纽结理论虽然是一个非常集中的数学活动，但在其他科学中有着广泛的分支，特别是在物理学、生物学和化学中。从古至今，结本身在编织和绳索的使用中都是物理性的。近年来，有人提出打结结构发生在原子核层次的量子场中，并且在量子引力理论和弦理论中出现了显着的结。在分子生物学中，DNA分子可能会打结，而避免打结的酶对于DNA的复制以及生命本身的维持至关重要。在化学中，分子和长链聚合物可以打结。这个暑期学校和会议的目的是检查这两个应用程序和理论的结在这个时候。 国际理论物理中心暑期学校和会议将来自世界各地、第三世界国家、妇女和少数民族的参与者聚集在一起，为研究人员和学生提供了一个分享最新想法和相互合作的机会。与会者将在世界各地传播在这次会议上获得的知识。所有强化短期课程均由专家授课，课堂笔记可在线获取。演讲包括杰出研究人员的全体会议，其中至少有8名妇女，以及其他与会者的简短发言。我们希望出版包含前沿研究论文和演讲稿的会议记录，这些论文和演讲稿将适合研究数学家，学生和具有其他精确科学背景的读者，包括生物学，化学，计算机科学和物理学。