Topology of Manifolds and Algebraic Varieties

该提案涉及拓扑和几何接口的工作。 Perelman 在将 Ricci 流应用于 3 流形拓扑方面取得了巨大进展。关于使用他的想法对封闭 3 流形进行完全分类（即建立瑟斯顿几何化猜想）仍然存在悬而未决的问题。这些问题围绕充分塌缩的 3 流形理论及其与最多 2 维的亚历山德罗夫空间的关系。整理这些问题将给出证明几何化猜想的完整、详细的论证。应该存在一个类似的充分塌陷的 4 流形理论以及这些物体与维度最多为 3 的 Alexandrov 空间的关系。开发这样的 4 流形理论本身就很重要，但对于应用 Ricci 流证明 4 维拓扑结果也可能至关重要。另一个研究领域是复曲面簇中 Calabi-Yau 3 折的镜像对称性。这里的想法是建立这些例子是在霍奇结构和单一性表示的相当微妙的变化水平上进行的。在过去几年中，拓扑学学科取得了令人兴奋的发展，本提案将探讨这些发展。这些涉及拓扑学与其他更具几何性和分析性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技术来建立庞加莱猜想，这是所有拓扑问题中最古老和最基本的问题，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、最美丽的应用之一。如何将相同的想法应用于所有 3 维空间，而不仅仅是最简单的 3 维球体，仍然存在许多问题。更具推测性的是，问题是如何扩展这些想法以提供对 4 维空间的新见解，我们对 4 维空间知之甚少，只是可能性比 3 维空间广泛得多。过去几年拓扑和几何的另一个主题是理解镜像对称性。这一原理是由非严格的物理论证推导出来的，与“经典数学”中通常看到的任何原理都不同。这是“量子数学”的某种版本。这个概念导致了拓扑学和几何学接口上数学的极其丰富的繁荣，因为数学家试图理解这个概念的特殊情况，精确地表述这个概念，并以数学方式建立它。

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.