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

结理论是非常特殊的拓扑学科：一个圆或圆的集合嵌入到三维空间的分类。这是一个经典的拓扑问题，也是一般布局问题的一个特例：理解空间X在另一个空间y中的嵌入。在过去的25年里，结理论和3流形拓扑领域有了令人兴奋的新发展。从Jones， Homflypt和Kauffman多项式，3-流形的量子不变量，到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，Khovanov将链接同源性作为Jones多项式的推广引入到链配合物的同源性中，Ozsvath和Szabo发展了Heegaard-Floer同源性，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。这些思想标志着结理论新时代的开始，其中包括与四维问题的关系以及与结理论相关的新代数拓扑形式的创建。一个值得注意的事实是，尽管结理论的问题形式非常特殊，但它包含了许多涉及数学和理论物理的思想和技术。这门学科与生物学、物理学、组合学、代数和计算理论有着重要的应用和联系。我们觉得这个主题的暑期学校在这个时候是非常合适的，我们努力在结理论及其应用的新发展的前沿。该项目有几个方面。它专注于研究结的理论。结理论，虽然是一个非常集中的数学活动，但在其他科学领域，特别是在物理学、生物学和化学领域，有着广泛的分支。从古代到现在，结本身在编织和绳索的使用中产生。近年来，有人提出在量子场中存在核水平的结结构，并且结在量子引力理论和弦理论中都有显著的存在。在分子生物学中，DNA分子会打结，避免打结的酶对DNA的复制至关重要，因此对维持生命本身至关重要。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在这个时候检查结的应用和理论。ICTP暑期学校和会议汇集了来自世界各地，第三世界国家，妇女，少数民族的参与者，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世界各地的与会者传播。所有强化短期课程将由专家授课，课堂讲稿将在网上提供。演讲包括杰出研究人员（包括至少8名女性）的全体会议，以及其他参与者的简短演讲。我们希望出版包含前沿研究论文和课堂讲稿的会议记录，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。

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.