Stability, derived categories, and mirror symmetry

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This project strives to understand problems in algebraic geometry through homological algebra, with inspiration from theoretical physics. Algebraic geometry is one of the oldest and richest fields of study in mathematics, dating back to the fifth century B.C. It is concerned with understanding geometric objects, called algebraic varieties, defined by the vanishing of polynomial functions, like circles or hyperbolas. Over the centuries, a vast array of technical tools have arisen for studying algebraic varieties. Relatively recently, homological algebra and mirror symmetry from physics have provided impressive insight. Even so, pressing questions persist. One in particular: What is the precise extent of the differences that can be detected between two varieties when using homological algebra or physics? This project seeks to elucidate this and related questions.The project focuses on a new tool for comparing derived categories: windows. Windows are a robust framework for establishing derived equivalences and semi-orthogonal decompositions, but, being very new, they are still developing. The project seeks to push the theory of windows by generalizing the framework and by reinterpreting the functors arising in terms of the geometry of compactifications and kernels. The project seeks to apply this new understanding to symplectic topology through mirror symmetry. Additionally, the project proposes to relate algebraic geometry to noncommutative algebra through the language of kernels and to use Hall algebras to study generation-style invariants of triangulated categories.