Conference on Topology, Geometry, and Physics; May 2006; New York, NY

拓扑、几何和物理会议；

• 批准号: 0540236
• 负责人: Robert Friedman
• 金额: $ 4万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-09-01 至 2007-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0540236&HistoricalAwards=false
• 关键词: Conference; Topology; Geometry; Physics; May

AbstractAward: DMS-0540236Principal Investigator: Robert D. Friedman, Peter S. OzsvathThis is a proposal for a four-day-long, broad conference atColumbia University on "Topology, Geometry, and Physics.'' Theconference will focus on the following major themes, and theirinterconnections: hyperbolic manifolds and geometrization gaugetheory, three- and four-manifolds, Hodge theory, and interactionsbetween mathematics and modern physics. Such a conference isparticularly timely, in view of the many breakthroughs in each ofthese subjects. In particular, the recent spectacular work ofPerelman on the Poincare conjecture and Thurston's geometrizationconjecture, as well as new developments in hyperbolic geometry,make this an especially opportune moment to survey the newlandscape of three-manifold theory. At the same time, newtechniques of gauge theory and new combinatorial methods havedeepened the current understanding of three- and four-manifoldsand it is time to take stock of these developments and theirrelation to other work in the field. Many of the new results ingeometric topology have been based upon ideas arising inmathematical physics and geometric analysis, and these kind ofinteractions will serve as a unifying theme for the conference.A fundamental problem in mathematics is to describe all possibleshapes. In the case of surfaces (two dimensional objects),possible shapes include a sphere (the surface of the earth, forexample), a torus (the surface of a tire) and more complicatedgeneralizations. A great deal of research has been concerned withdimension three, the dimension of space, and dimension four, thedimension of space-time. Because of its relevance tounderstanding our physical world, understanding all possibleshapes in these dimensions is particularlyimportant. Paradoxically, these cases are much harder tounderstand and to classify than higher-dimensional objects (whichin turn are much harder to visualize in any meaningful way). Thestudy of dimensions three and four has drawn on a rich variety ofmathematical and physical ideas. Recent work of a Russianmathematician, G. Perelman, seems to confirm one of the famousoutstanding conjectures of topology, the Poincare conjecture(which gives a complete characterization of the three dimensionalanalogue of a sphere), as well as a profound generalization ofthis conjecture, the geometrization conjecture of Thurston, whichgives in principle a scheme whereby one could describe allpossible three-dimensional shapes. A major goal of thisconference is to understand these new ideas, as well as otherrecent work in dimensions three and four, and to evaluate our newunderstanding of geometry and topology in these dimensions.

摘要奖：DMS-0540236主要研究者：Robert D.作者：Peter S.这是一个在哥伦比亚大学举行的为期四天的关于“拓扑学、几何学和物理学”的广泛会议的建议。会议将集中在以下主要主题，以及它们的相互联系：双曲流形和几何化规范理论，三，四流形，霍奇理论，数学和现代物理学之间的相互作用。鉴于这些主题都取得了许多突破，这样一次会议特别及时。特别是，最近壮观的工作ofPerelman的庞加莱猜想和瑟斯顿的geometrizationconjecture，以及新的发展，双曲几何，使这是一个特别有利的时刻，调查新景观的三流形理论。与此同时，规范理论的新技术和新的组合方法加深了目前对三维和四维流形的理解，现在是时候对这些发展及其与该领域其他工作的关系进行评估了。几何拓扑学中的许多新结果都是基于数学物理和几何分析中出现的思想，这些相互作用将成为会议的一个统一主题。数学中的一个基本问题是描述所有可能的形状。在表面（二维物体）的情况下，可能的形状包括球体（例如地球的表面），环面（轮胎的表面）和更复杂的概括。大量的研究都是关于三维空间和四维时空的。 由于它与理解我们的物理世界相关，理解这些维度中所有可能的形状特别重要。巧合的是，这些情况比高维对象更难理解和分类（高维对象更难以任何有意义的方式可视化）。三维和四维的研究已经从丰富的数学和物理思想中汲取了经验。俄国数学家G. 佩雷尔曼，似乎证实了著名的杰出的拓扑结构之一，庞加莱猜想（它给出了一个完整的特征的三维模拟的一个领域），以及一个深刻的推广这一猜想，几何化猜想的瑟斯顿，它在原则上给出了一个计划，使人们可以描述所有可能的三维形状。这次会议的一个主要目标是了解这些新的想法，以及其他最近在三维和四维的工作，并评估我们对这些维度的几何和拓扑学的新理解。