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

主要研究者：Robert D. Friedman, Peter S. ozsvath这是一个在哥伦比亚大学举行的为期四天的“拓扑、几何和物理”会议的提案。会议将集中讨论以下主要主题及其相互联系：双曲流形和几何化量规理论，三流形和四流形，霍奇理论，以及数学与现代物理之间的相互作用。鉴于这些学科都取得了许多突破，这样的会议尤其及时。特别是，佩雷尔曼最近在庞加莱猜想和瑟斯顿的几何化猜想上的惊人工作，以及双曲几何的新发展，使这成为一个特别合适的时机来调查三流形理论的新景观。同时，规范理论的新技术和新的组合方法加深了目前对三流形和四流形的理解，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。几何拓扑学的许多新结果都是基于数学物理和几何分析中产生的思想，这些相互作用将成为会议的统一主题。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（例如地球表面），环面（轮胎表面）和更复杂的概括。大量的研究关注于三维空间维度和四维时空维度。由于它与理解我们的物理世界的相关性，理解这些维度中所有可能的形状尤为重要。矛盾的是，这些情况比高维对象更难以理解和分类（高维对象更难以以任何有意义的方式可视化）。对三维和四维的研究借鉴了丰富多样的数学和物理思想。俄罗斯数学家G.佩雷尔曼最近的工作似乎证实了拓扑学中一个著名的杰出猜想，即庞加莱猜想（它给出了球体三维模拟的完整表征），以及对这一猜想的深刻推广，即瑟斯顿的几何化猜想，该猜想在原则上给出了一个可以描述所有可能的三维形状的方案。这次会议的一个主要目标是理解这些新思想，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的新理解。

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.