Topology of Manifolds and Algebraic Varieties

这一建议涉及拓扑和几何界面的工作。佩雷尔曼在将里奇流应用于3流形拓扑方面取得了重大进展。利用他的思想对闭3流形进行完全分类（即建立瑟斯顿的几何化猜想）仍然存在开放的问题。这些问题围绕着充分坍缩的3-流形理论及其与最多2维的Alexandrov空间的关系展开。整理这些问题将给出一个完整的、详细的论证来证明几何化猜想。应该有一个充分坍缩的4流形的类似理论，以及这些对象与不超过3维的Alexandrov空间的关系。为4流形发展这样的理论本身就很重要，但在应用里奇流证明4维拓扑结果时也可能至关重要。另一个研究领域是环形品种中Calabi-Yau 3褶皱的镜像对称性。这里的想法是为这些例子建立镜像对称，在相当微妙的Hodge结构和单一性表示的变化水平上。在过去的几年里，拓扑学的主题有了令人兴奋的发展，本提案将对此进行探讨。这涉及到拓扑学和其他几何和解析数学领域之间的相互作用。佩雷尔曼运用热流方程和几何学中的技巧建立了庞加莱猜想，这是所有拓扑问题中最古老和最基本的，是该学科的一个重要分水岭。这是偏微分方程在纯拓扑问题上最深刻、最美妙的应用之一。如何将同样的思想应用到所有的三维空间，而不仅仅是最简单的，也就是三维球体，还有很多问题。更有推测性的是，如何将这些想法扩展到四维空间，为我们所知甚少的四维空间提供新的见解，我们只知道四维空间的可能性比三维空间大得多。在过去的几年里，拓扑学和几何学的另一个主要主题是理解镜面对称。这个原理是由非严格的物理论证推导出来的，不同于通常在“经典数学”中看到的任何东西。这是某种版本的“量子数学”。这个概念在拓扑学和几何学的界面上带来了令人难以置信的数学繁荣，因为数学家们试图理解这个概念的特殊情况，精确地表述这个概念，并在数学上建立它。

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.