Cornell Topology Festival Support; 2003-2005

摘要奖：DMS-0244361首席研究员：彼得·J·卡恩康奈尔大学数学系的拓扑学/几何学小组请求提供资金，以继续支持康奈尔拓扑节三年，该组织自1963年以来每年都主办该节。艺术节为传播和发展重要的新成果提供了一个重要的舞台，这些成果来自代数、微分、几何拓扑学和相关学科领域的广泛领域。虽然计划延续多年来艺术节的广泛、跨学科的风格和有利于合作的形式，但也建议将艺术节延长一天，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，同时繁荣的两个子专业和跨学科的活动。将启动一个积极的外展计划，其中包括一些旅行支持，以扩大对艺术节的参与。在这项拨款下的第一次会议(2003年5月1日至4日)，我们将重点关注几何群论非常活跃的领域。拓扑节是在20世纪60年代构思和发起的，既是对新兴拓扑学领域的年度庆祝，也是美国东北部拓扑学家的年度会议/研讨会。虽然拓扑学在早期欧洲对空间和几何的数学研究中有很深的根基，但拓扑学是在20世纪中期才成熟的，美国数学家提供了大量的半官方发现。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关键因素。拓扑学是一个宽泛的学科，它与“离散”数学(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)重叠。事实上，当代数学的很大一部分已经受到拓扑学结果和方法的深刻影响。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(计量建模和模拟)、数理经济学(非光滑平衡理论)、物理学(规范场理论)以及数学中的邻近领域(例如几何群论)。虽然拓扑学早期的许多基本问题已经得到了回答，但随着科学的进步，新的问题不断地产生，重要的经典问题仍然存在。后者的一个例子是庞加莱猜想，在过去的拓扑节上，它一直是几个值得注意的演讲的主题，并且仍然是今天最重要的数学悬而未决的问题之一-克莱伊研究所的七个‘千禧年问题’之一-以及紧张研究的焦点。康奈尔大学的拓扑学家和几何学家为拓扑节所发挥的有益作用感到自豪，并将继续在这一充满活力的数学领域的当代发展中发挥作用。

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.