Mirror Constructions: Develop, Unify, Apply

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In this project, we research geometric problems inspired by string theory. In string theory, we view subatomic particles as strings, not points, requiring the universe to have six extra small dimensions called a Calabi-Yau shape. If we trace the string as it moves through time, it creates a (Riemann) surface. String theory has predicted amazing mathematics, which we, as mathematicians, prove rigorously.We are mainly focussed on studying shapes that can be viewed as the solution to a set of polynomial equations. Chosen with the correct data, such a system of equations can be used to define a Calabi-Yau shape. String theory predicts a duality that states that, for any Calabi-Yau space, there exists another space called the mirror. Various physical and geometric data between these two shapes is exchanged, creating a relationship that has come to be known as mirror symmetry. A key problem in this field is how one, given the Calabi-Yau space, finds the mirror space that is related to it. Once an explicit construction is developed, we then can check if a mirror relationship holds. There are various constructions in the literature with varying degrees of evidence of mirror symmetry; however, they often disagree! We aim in this project to deal with this discrepancy, unifying their approaches. In the same vein, we aim to potentially create new Calabi-Yau varieties while also giving their mirror shape, adding to the library of mirror pairs that currently exist.While Calabi-Yau spaces are often very difficult to visualize, they often have algebraic descriptions that are easy to study. In this project, we often will deform the Calabi-Yau shape so much that it is no longer even a Calabi-Yau space but some easier algebraic structure, known in the physics literature as a Landau-Ginzburg model. By proving relations between Landau-Ginzburg models, we will often find relations between Calabi-Yau shapes themselves. Thus, we will be able to relate various constructions algebraically in order to create a better overview of mirror proposals. Indeed, this explains the discrepancy above between different constructions for mirrors in the literature.In addition, we will study the algebraic relations to Landau-Ginzburg models in order to create new relations between Fano manifolds. While there is a large project regarding classification of Fano manifolds in low dimension, they often have the same interesting or intrinsic piece of algebraic structure, known as a (fractional) Calabi-Yau category. We aim to apply our intuition from unifying constructions in order to find relations between this fundamental data in order to streamline the relations between potential Fano manifolds. Lastly, we apply our understanding of the geometry of various Calabi-Yau spaces to computational number theory. The one-dimensional case of a Calabi-Yau shape, the elliptic curve, has played a leading role in cryptography in the last few decades; however, there have been recent proposals that have led to needing more understanding of higher dimensions. By interacting with computational number theorists, we will isolate fundamental Calabi-Yau shapes that exhibit interesting explicit number-theoretic phenomena, leading to applications for L-series.