Exotic smooth structures on 4-manifolds

我提议的研究将探索称为四维流形的四维空间。四维流形的几何学在物理学和工程学中有许多重要的应用，因为四维流形可以作为我们物理世界时空的有效数学模型。为了让我们在给定的4-流形上执行多变量微积分，我们需要在4-流形上选择一个光滑结构，即，我们必须选择四维流形上的哪些函数是可微的。对于一个具有足够复杂度的四维流形，我们现在知道在它上面有无穷多个不同的光滑结构，这样的光滑结构的无穷多个集合被称为奇异光滑结构。奇异光滑结构的存在性意味着许多4-流形上的偏微分方程组对于一个光滑结构可能没有解，但对于另一个光滑结构可能有多个解。因此，我们宇宙的数学模型的某些大尺度行为将主要取决于奇异光滑结构的分类。我的研究项目的主要重点是在更广泛的一类4-流形上构造奇异光滑结构，重点是具有很少或没有复杂性的4-流形。由于复杂性很小的四维流形最常用作数学模型，因此我的项目成果将在各种环境中找到应用，包括但不限于宇宙学，相对论和量子场论。

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.