Topology of Manifolds and Algebraic Varieties

这一建议涉及拓扑学和几何学的接口工作。Perelman在将Ricci流应用于三维流形拓扑方面取得了很大的进展。关于使用他的思想来完全分类闭三维流形（即，Thurston几何化猜想（Thurston's Geometriization Conjecture）这些问题围绕着充分坍缩的三维流形理论及其与维度至多为2的亚历山德罗夫空间的关系。将这些问题整理出来，将给出一个完整的，详细的论证证明几何化猜想。应该有一个类似的理论充分塌陷4流形和这些对象的关系，以亚历山德罗夫空间的维度最多3。发展这样一个4维流形的理论本身就很重要，但在应用里奇流来证明4维拓扑结果时也可能是至关重要的。研究的另一个领域是镜像对称的卡-丘3倍复曲面品种。这里的想法是建立镜像对称的这些例子在相当微妙的水平变化的霍奇结构和monodromy representations.There一直是令人兴奋的发展，在过去的几年中，拓扑学的主题，这一建议将探讨。这些涉及到拓扑学和其他更多的几何和分析数学领域之间的相互作用。佩雷尔曼的应用热型流动方程和技术从几何建立庞加莱猜想，最古老和最基本的所有拓扑问题，是一个重大的分水岭时刻的主题。这是一个最深刻和最美丽的应用偏微分方程的一个纯粹的拓扑问题。如何将同样的思想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。更进一步的问题是，如何扩展这些想法，为四维空间提供新的见解，我们对四维空间知之甚少，只知道可能性比三维空间大得多。过去几年拓扑学和几何学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，与通常在“经典数学”中看到的任何东西都不一样。这是量子数学的一个版本。这一概念导致了数学在拓扑学和几何学的界面上令人难以置信的丰富繁荣，因为数学家试图理解这一概念的特殊情况，精确地表达这一概念，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.