Mirror Symmetry for Fibrations and Degenerations

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Mirror symmetry, as a discipline, has its roots in theoretical physics and string theory. The core idea of string theory is that subatomic particles are tiny loops of string, instead of points. The subatomic physics that we observe then arises as these loops of string vibrate, move about, and interact with each other. However, to produce the physical properties that we observe in our universe, the strings need more space to move than is afforded to them by our usual four dimensions (3 space and 1 time). To solve this problem, string theory postulates that the universe should have six tiny extra dimensions, which are coiled up together into a shape called a "Calabi-Yau manifold". There are many different Calabi-Yau manifolds, and which one we use in string theory is important. Just as changing the speed of light would fundamentally alter the physics of our universe, so too should changing the Calabi-Yau manifold. However, early in the development of string theory, physicists noticed a curious anomaly: every Calabi-Yau manifold seems to have a partner Calabi-Yau manifold, which gives identical physical predictions when passed through the string theory machinery.This observed pairing-up of Calabi-Yau manifolds was the first known example of mirror symmetry. Mathematically, mirror symmetry can be thought of as the idea that many geometric objects (such as Calabi-Yau manifolds) have a "mirror partner": a second geometric object whose properties are closely related to the first.This is a tremendously powerful mathematical tool. Often, difficult mathematical questions about a geometric object can be translated, through mirror symmetry, into much simpler questions about its mirror partner. However, there is a fundamental problem that restricts the use of this in practice: given a geometric object, we usually have no idea how to construct a mirror partner for it! Attempts to solve this problem have led to a number of ad-hoc definitions of mirror partners, each of which works for some types of geometric objects and completely fails for others. This leads to the second fundamental problem of mirror symmetry: is there a single overarching theory that combines all of the different formulations into one consistent framework?This proposal aims to address this second question by showing that two of the most frequently used formulations of mirror symmetry are actually parts of one bigger picture. The two formulations in question are "Calabi-Yau mirror symmetry", which is the original formulation for Calabi-Yau manifolds as described above, and the "Fano/LG correspondence", which states that the mirror partner of a geometric object called a "Fano manifold" is a "Landau-Ginzburg (LG) model".A powerful application of this theory, that will also be studied as part of this proposal, is to the construction of new mirror pairs of Calabi-Yau manifolds. To do this, one starts with Fano manifolds and their mirror partner LG models; many examples of such pairs are known. Using the theory developed in this proposal, one may glue together Fano manifolds to get a Calabi-Yau manifold, and glue together their mirror partner LG models to get a second Calabi-Yau manifold, such that the two Calabi-Yau manifolds obtained are mirror partners.