Conference at MSRI in Low Dimensional-Topology

9727594 Casson这项奖励将提供演讲者和与会者参加1998年5月22-26日在加利福尼亚州伯克利数学科学研究中心举行的低维拓扑学会议的部分旅费和生活费。会议的目的是将所有低维拓扑学领域的研究人员聚集在一起，使他们能够对该领域的发展有一个全面的看法。近年来，三维和四维流形和纽结理论的研究取得了令人振奋的新进展。将这些领域的研究人员聚集在一起，鼓励思想和技术的交叉培养，让研究生接触到这些想法，并鼓励新的合作。这次会议是在M.S.R.I.一个密切相关的为期一年的项目结束一年后举行的。因此，很自然地就会回顾在接下来的一年里这一领域的进展是如何得到巩固的，而会议的另一个好处是向更广泛的数学界展示这一进展。大约150名数学家参加了会议，其中50多人是研究生。一些讲座是专门为这些学生量身定做的。拓扑学解决了那些只依赖于连续性的几何属性，即一个点靠近另一个点的概念，而忽略了高中几何交易中的存量，如距离、角度、面积、体积等量的测量。平面的拓扑学在通俗的科学文献中通常被称为橡胶片几何，这抓住了这门学科的基本精神，却没有给出拓扑学在他们的科学中可能使用的技术的任何暗示。他们面临着一些非常基本但难以捉摸的属性，那些在僵化被驱除后仍然存在的属性，在过去的一个世纪里，人们逐渐组装了相当多的武器库来处理这些属性。低维拓扑，主要是三维和四维几何对象的拓扑，本身就有特殊的问题，因为虽然我们有一种直觉，我们应该能够更容易地理解这些对象，就像我们生活在三维空间和四维时空中一样，但实际上，为拓扑学而开发的许多代数机械在较低的维度上失败了。近年来，在这方面也取得了一些显著的进展，这也是M.S.R.I.in May会议的中心主题。同样值得注意的是，低维拓扑学已经发展得非常有用，例如，在将纽结理论应用于聚合物化学和DNA研究中。随着这一领域的继续发展，它所处理的对象的直接性很可能会导致其进一步的技术进步，在现实世界中找到同样自然的应用。

9727594Casson This award will provide partial reimbursement for travel andsubsistence of speakers and participants in a conference onlow-dimensional topology that took place at the Mathematical SciencesResearch Center, Berkeley, California, May 22-26, 1998. The purpose ofthe conference was to bring together researchers working in all strands oflow-dimensional topology and to enable them to gain an overall perspectiveon developments in the field. Recent years have seen exciting newdevelopments in the study of 3- and 4-dimensional manifolds and knottheory. Bringing researchers in these fields together encouraged across-fertilization of ideas and techniques, exposed graduate studentsto these ideas, and encouraged new collaborations. The conference tookplace one year after the end of a closely related year-long program atM.S.R.I. Thus it was natural to review how progress had beenconsolidated in this area over the ensuing year, and an added benefit ofthe conference was to expose this progress to the wider mathematicalcommunity. Approximately 150 mathematicians attended the conference,more than 50 of them graduate students. Some talks were tailoredspecifically for these students. Topology addresses those geometric properties that depend only oncontinuity, i.e., on the notion of one point being close to another, butdisregards the stock in trade of high school geometry, measurement ofquantities such as distance, angle, area, volume, and so forth. Topologyof the plane is often referred to in popular scientific literature asrubber sheet geometry, and this captures the essential spirit of thediscipline without giving so much as a hint of the techniques topologistsmight use in their science. They confront some very basic but elusiveproperties, those that remain after rigidity has been exorcised, and thepast century has seen the gradual assembly of a considerable arsenal ofweapons for dealing with these properties. Low-dimensional topology,mainly the topology of 3- and 4-dimensional geometric objects, presentsspecial problems of its own, for while we have the intuition that weshould be able to apprehend these object more readily, living as we do in3-dimensional space and 4-dimensional space-time, in fact much of thealgebraic machinery that has been developed for topology fails in theselower dimensions. Recent years have seen some remarkable advances on thisfront too, and this was the central theme of the conference at M.S.R.I. inMay. It is also noteworthy that low-dimensional topology has becomesufficiently highly developed to be useful, for example, in applicationsof knot theory to polymer chemistry and to the study of DNA. As this fieldcontinues to progress, the immediacy of the objects with which it deals islikely to lead its further technical advances to find equally naturalapplications in the real world.***