Homological Algebra of Landau-Ginzburg Mirror Symmetry

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This is a research project to establish algebraic and geometric results inspired by a duality originally from string theory.Right before the turn of the century, theoretical physics provided an insight to geometry that led to many modern successes in geometric research. They discovered a duality in string theory had implications and applications to the study of higher dimensional geometry, making answers to classical questions about the geometry of certain six-dimensional spaces accessible. Roughly speaking, for (classical) string theory to provide a potential physical theory for the universe, it requires the universe to be 10-dimensional. Four of these dimensions are the standard 3 space dimensions and one time dimension we experience in our lives, and the other six are a so-called Calabi-Yau manifold. It is still unclear how many Calabi-Yau manifolds there are, and we study them in many different ways, but string theory has given us a deep connection between geometric research disciplines. In particular, a duality in string theory states that each Calabi-Yau manifold has a "mirror" which is another Calabi-Yau manifold so that various geometric and physical properties of one are encapsulated in other geometric and physical properties of its mirror. This phenomenon in mathematics is now known as mirror symmetry. In particular, hard computations and computational open questions in symplectic geometry associated to a Calabi-Yau manifold were now encoded in the algebraic geometry of its mirror. At the onset of mirror symmetry, these algebro-geometric computations were much easier and then they were then used as a guiding principle for what we aim to prove in symplectic geometry. This made century-old problems in enumerative geometry achievable. In 1994, the Fields Medallist Kontsevich provided a conjectural but fully mathematical version of mirror symmetry, encoding the symplectic geometry in what is called a Fukaya category and the algebraic geometry in a derived category of coherent sheaves. This provided a robust formulation in algebra of this physical and geometric phenomenon.Throughout the past three decades, mirror symmetry has expanded and it is now seen that mirror symmetry is not just a relationship amongst Calabi-Yau manifolds, but many more geometric spaces (e.g., Fano manifolds, log Calabi-Yau varieties). However, it has also been extended to the study of singularities. Interestingly, one can model the geometry of certain spaces by constructing a function so that the function is singular along the original space. Then one can deform this model and still obtain a physical model for string theory. This is an example of a Landau-Ginzburg model. Mirror symmetry has been established for Landau-Ginzburg models in a few cases, and it has been shown to be powerful in the study of classical higher-dimensional shapes such as Calabi-Yau manifolds. However, there are still foundational issues to be handled in the study of mirror symmetry for Landau-Ginzburg models. Ideally, we would like to prove a form of Kontsevich's conjecture for Landau-Ginzburg models, but before we do so in general, we will need to understand the algebro-geometric aspects of Landau-Ginzburg models. This project aims to better understand this categorical point of view for Landau-Ginzburg models, proving various structural results on their analogue of the derived category of coherent sheaves above, known as the (matrix) factorisation category.