Open Mirror Geometry for Landau-Ginzburg Models

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This fellowship researches geometric problems made accessible by string theory. In string theory, one views subatomic particles as strings, not points, requiring the universe to have six extra small dimensions in the form of what is known as a Calabi-Yau shape. If we trace the string as it moves through time, it creates a (Riemann) surface. String theory has pointed out mathematical structure that we initially did not see. Now, mathematics informed by string theory has created great advances that has pushed the boundaries of geometry, algebra and even string theory itself.The most clear way in which string theory has revolutionised modern geometry is in enumerative geometry. An example of an enumerative problem is "How many lines in the cartesian plane go through two given points?" The answer is something we have known since secondary school: there's a unique straight line between two points. We now ask: "How many Riemann surfaces / strings (of a given degree or genus) are in a Calabi-Yau shape?" This is a classical problem in geometry, going back in some form to the 19th century. Such questions were the basis to Hilbert's 15th problem posed in 1900. Answering enumerative problems like this one helps us understand higher-dimensional spaces, a key problem for geometers. Here, we use modern ideas to tackle problems over a century old, while also studying contemporary variants.Today, we count Riemann surfaces by using dualities in string theory to encode the counts into multivariate integration. Such a relation is the manifestation of a field called mirror symmetry. This duality exchanges the data of two different Calabi-Yau shapes using different types of geometry: (1) the enumerative geometry that sits squarely in the field of symplectic geometry and (2) the multivariate integration which is placed in the field of algebraic geometry. The two Calabi-Yau shapes that have this exchange of data are called mirrors. Mirror symmetry has been one of the key catalysts of modern geometry for the past thirty years, and is only gaining momentum.A key question in mirror symmetry is: given a Calabi-Yau shape, how do I construct the mirror? More broadly, one can ask how to construct the mirror for any symplectic manifold and its deformations. In the past 10 years, there has been work in trying to understand how mirror symmetry works for all deformations of a given Calabi-Yau shape. Roughly speaking, when one deforms a Calabi-Yau shape too hard, one ends up with not a space anymore, but a complex-valued function known as a Landau-Ginzburg model. The geometry of the Calabi-Yau shape is now encapsulated in this function, where it is easier to compute. The analogous theory for counting Riemann surfaces for Landau-Ginzburg models, known as FJRW theory or quantum singularity theory, was developed in 2013; however, there is no systematic way in any large generality for how one can construct the mirror to a Landau-Ginzburg model.This fellowship aims to solve the key question above for Landau-Ginzburg models, providing a way to construct the mirror to a Landau-Ginzburg model directly. In effect, this will provide a more 'global' approach to constructing mirrors, allowing for one to study deformations of symplectic spaces more effectively.