Exotic smooth structures on 4-manifolds

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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.