Topology of Manifolds and Algebraic Varieties

流形拓扑和代数簇

• 批准号: 0706815
• 负责人: John Morgan
• 金额: $ 33.24万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706815&HistoricalAwards=false
• 关键词: Topology; Manifolds; Algebraic; Varieties

This proposal concerns work at the interface of topology and geometry. Perelman has made great advances in applying Ricci flow to the topology of 3-manifolds. There remain open questions about using his ideas to completely classify closed 3-manifolds (i.e., establish Thurston's Geometrization Conjecture). These questions revolve around the theory of sufficiently collapsed 3-manifolds and their relationship to Alexandrov spaces of dimension at most 2. Sorting these issues out will give a complete, detailed argument proving the Geometrization Conjecture. There should be an analogous theory of sufficiently collapsed 4-manifolds and the relation of these objects to Alexandrov spaces of dimension at most 3. Developing such a theory for 4-manifolds will be important in its own right, but may also be crucial in applying Ricci flow to prove topological results in dimension 4. The other area of study is mirror symmetry for Calabi-Yau 3-folds in toric varieties.Here the idea is to establish mirror symmetry for these examples at the fairly delicate level of variations of Hodge structure and monodromy representations.There have been exciting developments in the subject of topology in the last few years which this proposal will explore. These involve the interplay between topology and other, more geometric and analytic, areas of mathematics. Perelman's application of a heat-type flow equation and techniques from geometry to establish the Poincare Conjecture, the oldest and most fundamental of all topological questions, is a major watershed moment in the subject. This is one of the deepest and most beautiful applications of partial differential equations to a purely topological problem ever. There remain many questions of how to apply the same ideas to all 3-dimensional spaces, not just simplest, which is the 3-dimensional sphere. More speculatively, there is the question of how to extend these ideas to provide new insights into 4-dimensional spaces, about which we know little, except that the possibilities are much wider than in dimension 3. Another main theme in topology and geometry over the past few years is to make sense of the mirror symmetry. This principle is derived by non-rigorous physics arguments and is unlike anything normally seen in `classical mathematics.' It is some version of `quantum mathematics.' This notion has led to an incredibly rich flourishing of mathematics on the interface of topology and geometry as mathematicians attempt to understand special cases of this notion, to formulate this notion precisely, and to establish it mathematically.

这项建议涉及到拓扑学和几何学的交界处的工作。Perelman在将Ricci流应用于三维流形的拓扑学方面取得了很大进展。关于用他的思想对闭3-流形进行完全分类(即建立瑟斯顿的几何猜想)，仍有许多悬而未决的问题。这些问题围绕着充分折叠的3-流形理论及其与最多2维Alexandrov空间的关系。对这些问题的梳理将给出一个完整的、详细的论证来证明几何猜想。应该有一个充分折叠的4-流形的类似理论，以及这些对象与最多3维Alexandrov空间的关系。发展这样一个关于4-流形的理论本身将是重要的，但在应用Ricci流来证明4维的拓扑结果方面也可能是至关重要的。另一个研究领域是环面变种中Calabi-Yau 3-折叠的镜像对称性。这里的想法是在Hodge结构和一元表示的相当微妙的变化水平上为这些例子建立镜像对称性。在过去的几年里，这个提议将探索的拓扑学学科已经有了令人兴奋的发展。这些问题涉及拓扑学和其他更几何、更解析的数学领域之间的相互作用。佩雷尔曼应用热型流动方程和几何学中的技术建立了庞加莱猜想，这是所有拓扑问题中最古老和最基本的，是该学科的一个主要分水岭时刻。这是偏微分方程在纯拓扑问题中最深刻、最美丽的应用之一。如何将同样的想法应用于所有的三维空间，而不仅仅是最简单的三维球体，仍然存在许多问题。更具推测性的是，还有一个问题是如何将这些想法扩展到四维空间，为我们提供对四维空间的新见解，我们对四维空间知之甚少，除了可能性比三维空间广泛得多。过去几年拓扑学和几何学的另一个主要主题是理解镜像对称。这一原理是从非严格的物理论证中推导出来的，不同于在“经典数学”中通常看到的任何东西。这是某种版本的“量子数学”。随着数学家试图理解这一概念的特殊情况，精确地表述这一概念，并以数学的方式建立它，这一概念导致了拓扑和几何界面上令人难以置信的丰富的数学繁荣。