Exotic smooth structures on 4-manifolds

4 歧管上的奇异光滑结构

• 批准号: 261491-2012
• 负责人: Park, Doug
• 金额: $ 2.19万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=525246
• 关键词: Exotic; smooth; structures; manifolds

My proposed research will explore 4-dimensional spaces called 4-manifolds. The geometry of 4-manifolds has many important applications in physics and engineering since 4-manifolds can serve as effective mathematical models for our physical world of space-time. In order for us to perform multivariable calculus on a given 4-manifold, we need to choose a smooth structure on the 4-manifold, i.e., we must choose which functions on the 4-manifold are differentiable. For a 4-manifold with enough complexity, we now know that there are infinitely many distinct smooth structures on it. Such an infinite collection of smooth structures are called exotic smooth structures. The existence of exotic smooth structures implies that many systems of partial differential equations on a 4-manifold may have no solution with respect to one smooth structure but have multiple solutions with respect to another smooth structure. Hence certain large-scale behavior of mathematical models for our universe will depend essentially on the classification of exotic smooth structures. The main focus of my research project is to construct exotic smooth structures on a wider class of 4-manifolds, with an emphasis on the 4-manifolds with little or no complexity. Since the 4-manifolds with little complexity are most frequently used as mathematical models, the outcome of my project will find applications in a variety of settings including, but not limited to, cosmology, relativity, and quantum field theory.

我提议的研究将探索称为4-流形的4维空间。4-流形的几何在物理和工程上有许多重要的应用，因为4-流形可以作为我们时空物理世界的有效的数学模型。为了在给定的4-流形上进行多元微积分，我们需要在4-流形上选择一个光滑的结构，即我们必须选择4-流形上的哪些函数是可微的。对于一个具有足够复杂性的4-流形，我们现在知道它上有无限多个不同的光滑结构。这种光滑结构的无限集合被称为奇异光滑结构。奇异光滑结构的存在意味着四维流形上的许多偏微分方程组关于一个光滑结构可能没有解，但关于另一个光滑结构有多个解。因此，我们宇宙数学模型的某些大规模行为本质上将取决于奇异光滑结构的分类。我的研究项目的重点是在更广泛的4-流形上构造奇异的光滑结构，重点是复杂性很低或没有复杂性的4-流形。由于复杂性不高的4维流形最常被用作数学模型，我的项目成果将在各种环境中得到应用，包括但不限于宇宙学、相对论和量子场论。