Windows and Mirror Symmetry

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The set of solutions to polynomial equations form geometric shapes (e.g. lines, planes, curves). Algebraic geometry studies the connection between the solutions to these polynomial equations and the geometry of these shapes. Symplectic geometry also studies geometric shapes but disregards any algebraic natural they may have, retaining instead a way to measure the area inside. In the early 1990s, high-energy theoretical physicists found that questions in symplectic geometry can be answered using algebraic geometry and vice versa. This connection between these two areas of mathematics is called "mirror symmetry". In 1994, during his Fields Medal address, Maxim Kontsevich proposed a vast mathematical framework for mirror symmetry, connecting algebraic geometry and symplectic geometry through category theory, a modern mathematical language which can be used to describe most mathematical topics. Kontsevich's formulation of mirror symmetry was completely revolutionary and without any precedent in mathematics. In the last 25 years, it has burgeoned into a massive field of study attracting a diverse international field of researchers from algebraic geometry, symplectic geometry, and high-energy theoretical physics. As an essential part of his program, Kontsevich distilled the algebro-geometric side of mirror symmetry into a mathematical concept known as the derived category. This research proposal aims to provide a fundamental description of derived categories, especially as they appear in models from high-energy theoretical physics. It decomposes derived categories into simpler pieces, makes comparisons between them, and aims to prove some of the biggest questions remaining about their behavior. It also constructs new versions of mirror symmetry within Kontsevich's framework. Finally, it aims to apply the resulting theory to solve classical problems in algebraic geometry.