Cornell Topology Festival Support; 2003-2005

AbstractAward：DMS-0244361首席研究员：康奈尔大学数学系的Peter J. Kahnell拓扑/几何小组请求资助继续支持康奈尔拓扑节三年，该小组自1963年以来每年都主办该节日。该节日提供了一个重要的tarena的传播和发展的重要，新的成果，从广泛的领域内的领域ofalgebraic，微分，几何拓扑结构，和alliedsubjects。 虽然计划继续广泛的，跨学科的风味和合作友好的形式，这已经成为电影节多年来的特点，但也建议将电影节延长一天，以适应特定的年度重点领域。通过这种方式，艺术节可以响应这个时代的主导趋势：即，同时繁荣的亚专业和跨学科活动。一个积极的外展计划，其中包括一些旅行支持，将启动，以扩大参与电影节。对于我们的第一次会议根据这项补助金（2003年5月1日至4日），我们将有作为重点领域的非常活跃的领域的几何群论。拓扑节的构想和发起于20世纪60年代都作为一个新兴的领域topologyandas一个年度会议/研讨会的拓扑学家在美国东南部。 虽然拓扑学在欧洲早期的空间和几何数学研究中有着深厚的根基，但直到世纪中期，随着美国数学家提供了大量的数学发现，拓扑学才成熟起来。美国的拓扑学一直是美国在数学领域处于世界领先地位的一个关键因素。 拓扑学是一门广泛的学科，它与“离散”数学的领域重叠（例如，代数，组合数学），以及“连续”或“光滑”数学（例如，微分几何、动力系统、数学物理）。 事实上，当代数学的很大一部分已经深刻地受到拓扑结果和方法的影响。拓扑学作为一门基础学科，其应用十分广泛.一些例子是：遗传学（DNA分子打结的研究），神经科学和机器人学（动力系统建模），计算机科学（几何建模和仿真），数学经济学（不光滑平衡理论），物理学（规范场理论）和数学中的邻近领域（例如，几何群论）。 虽然许多基本的问题，动力早期的拓扑学已经回答，新的问题是不断产生的科学向前发展，和重要的经典问题仍然存在。 后者的一个例子是庞加莱猜想，它一直是过去拓扑学论坛上几次著名会谈的主题，并且仍然是今天数学中最重要的未解决问题之一--克莱研究所的七个“千年问题”之一--并且是紧张研究的焦点。拓扑学家和几何学家在康奈尔大学感到自豪的有用的作用，拓扑节已经发挥，并将继续发挥在当代发展的这一动态领域的数学。

AbstractAward: DMS-0244361Principal Investigator: Peter J. KahnThe topology/geometry group in the Department of Mathematics atCornell University requests funding to continue support for threeyears for the Cornell Topology Festival, which this group hashosted each year since 1963. The Festival provides a significantarena for the dissemination and development of important, newresults from a wide array of areas within the realms ofalgebraic, differential, and geometric topology, and alliedsubjects. While it is planned to continue the broad,interdisciplinary flavor and collaboration-friendly format thathave characterized the Festival over the years, it is alsoproposed to extend the Festival by a day to accommodate specificannual areas of emphasis. In this way the Festival can beresponsive to the dominant trends of this era: namely, thesimultaneous flourishing of both subspecializations andcross-disciplinary activity. An active outreach program, whichincludes some travel support, will be initiated to broadenparticipation in the Festival. For our initial conference underthis grant (May 1-4, 2003), we will have as area of emphasis thevery active field of geometric group theory.The Topology Festival was conceived and initiated in the 1960'sboth as an annual celebration of the burgeoning field of topologyand as an annual conference/workshop for topologists in thenortheastern United States. While topology has deep roots inearly European mathematical investigations of space and geometry,it is in the mid-twentieth century that topology came of age,with American mathematicians providing a large array of seminaldiscoveries. American topology has been a key element inAmerica's world leadership in mathematics. Topology is a broadsubject, which overlaps with areas of `discrete' mathematics(e.g., algebra, combinatorics), as well as with `continuous' or`smooth' mathematics (e.g., differential geometry, dynamicalsystems, mathematical physics). Indeed, a large part ofcontemporary mathematics has been profoundly affected bytopological results and methods. As a foundational subject, theapplications of topology are very widespread. A few examples are:genetics (the study of knotting of DNA molecules), neuroscienceand robotics (dynamical systems modeling), computer science ( ingeometric modeling and simulation), mathematical economics (insmooth equilibrium theory), physics (the theory of gauge fields),and neighboring fields in mathematics (e.g., geometric grouptheory). Although many of the basic questions that powered earlyefforts in topology have been answered, new problems areconstantly being generated as science moves forward, andimportant classical problems remain. An example of the latter isthe Poincare Conjecture, which has been the subject of severalnotable talks at past Topology Festivals and remains one of thepremier unsolved problems of mathematics today---one of the ClayInstitute's seven `millenium problems'--- and the focus ofintense research. The topologists and geometers at Cornell areproud of the useful role that the Topology Festival has playedand will continue to play in the contemporary development of thisdynamic area of mathematics.