Mathematical Sciences: Index Theory, Coarse Geometry and Topology of Manifolds

福克斯 NSF/CBMS区域研究会议提案#9414820摘要： 数学的一个基本目标是找出所研究对象的基本特征，即把这些对象与其他对象区别开来的特征。例如，球与甜甜圈的区别在于甜甜圈有一个孔，而球没有。美国国家科学基金会的CBMS计划，RIndex理论，粗几何，拓扑流形，S将给出一个概述目前的工作，在寻找一类几何对象的不变量。 本研究中研究的不变量类型与甜甜圈/球示例中的不变量类型在精神上相似，但要复杂得多，也更有用。 已经有迹象表明，这项研究可能在生物技术的爆炸性领域有深远的应用。 例如，指数理论为生物化学家提供了描述和分类生命大分子在空间中扭曲和转动的各种方式的工具，这是一个困难而极其重要的问题。 牛津大学的John Roe将在这个多方面的数学领域做10场演讲。这些讲座有几个目的。 他们将提出新的和困难的想法，融合数学的几个领域，包括拓扑，几何和分析。 他们还将向研究生介绍开创性和基础数学研究。此外，约翰罗伊，一个天才作家对数学，将产生一个专着这些讲座，这将证明是一个宝贵的资源，为那些出席，并为广大的数学界。

Fox NSF/CBMS Regional Research Conference Proposal #9414820 ABSTRACT: A fundamental goal of mathematics is to find the RinvariantsS of objects studied---the characteristic signatures that distinguish these objects from others. For example, a ball is distinguished from a donut by the fact that a donut has one hole while the ball has none. The NSF CBMS program, RIndex Theory, Coarse Geometry, and Topology of Manifolds, S will give an overview of current work in finding invariants for a class of geometric objects. The type of invariants studied in this research are similar in spirit to those in the donut/ball example, though vastly more complicated and useful. There is already a hint that this research could have profound applications in the exploding field of biotechnology. For example, index theory gives biochemists the tools to describe and classify the various ways that the macro-molecules of life twist and turn in space, a difficult and extremely important problem. John Roe of Oxford University will give ten lectures on this multifaceted area of mathematics. The lectures will serve several purposes. They will present new and difficult ideas that blend several areas of mathematics, including topology, geometry and analysis. They will also introduce graduate students to ground-breaking and fundamental mathematical research. Additionally, John Roe, a gifted writer on mathematics, will produce a monograph on these lectures, which will prove a valuable resource for those attending, and for the mathematics community at large.