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

摘要奖：DMS-0540236主要研究人员：罗伯特·D·弗里德曼，彼得·S·奥兹瓦特这是一项在哥伦比亚大学举行为期四天的广泛会议的建议，主题是“拓扑学、几何学和物理学”。会议将集中讨论以下主要主题及其相互联系：双曲流形和几何规范理论，三和四流形，霍奇理论，以及数学和现代物理之间的相互作用。鉴于这些主题都取得了许多突破，这样的会议特别及时。特别是，佩雷尔曼最近关于庞加莱猜想和瑟斯顿几何化猜想的壮观工作，以及双曲几何的新发展，使现在成为考察三维流形理论新格局的一个特别合适的时刻。同时，规范理论的新技术和新的组合方法加深了对三、四流形的理解，现在是时候评估这些发展及其与该领域其他工作的关系了。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，这种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。已经有大量的研究涉及到空间的维度三和时空的维度四。因为它与理解我们的物理世界有关，所以理解这些维度上的所有可能的形状尤其重要。矛盾的是，这些情况比更高维度的物体更难理解和分类(反过来，更难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著名的杰出拓扑学猜想之一庞加莱猜想(它给出了球体的三维模拟的完整特征)，以及这个猜想的深刻推广，瑟斯顿的几何化猜想，它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了解这些新的概念，以及其他最近在三维和四维领域的工作，并评估我们对这些维度上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.