Mirror Constructions: Develop, Unify, Apply

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.

Fundamental research in pure mathematics often takes a long time in order to directly impact society at large. It tends to have a consequence that is almost impossible to foresee when the pure theorem is originally stated. Sometimes fundamental research that has a strong academic impact will lead to groundbreaking societal impact. For example, the usage of the arithmetic of elliptic curves in modern cryptography and the application of tropical geometry by the Bank of England in the 2008 economic crisis were unforeseen implications of pure mathematical research in nearby fields.

It is important to create a healthy mathematical dialogue in the advent for when such an application appears. In order to maximise the possibility of an application to be more immediate, a strong dissemination strategy is required. This dissemination strategy is addressed in the "Pathways to Impact."

By maintaining and further developing a world-class mathematical community in the United Kingdom, we attract the internationally-leading students at the undergraduate and postgraduate levels to our top tier mathematical programmes. These students then go on to lead the scientific industry community in the United Kingdom as well as academic institutions. The short term impact of this project is educational, helping build on the world-leading status of the mirror symmetry community in the UK, in order to develop the next generation of top tier researchers and leaders in industry. Before the start of the grant, I will have already started to help with the image of the UK on an international level. In July 2019, I will be a mentor in the Undergraduate Workshop on Landau-Ginzburg models at the Fields Institute at the University of Toronto. This builds the publicity of interesting research to students that could then help to recruit internationally leading students to mirror symmetry and to potential PhD places in the UK.

During my time at the University of Pennsylvania, I helped run a yearlong Emerging Scholars Program to recruit underrepresented students into STEM fields. Underrepresented groups have traditionally not studied in STEM fields, which leaves an unfortunately untapped resource of talent that the UK can explore. I have requested funds in order to contribute in University of Birmingham's already robust outreach initiative, creating new programming for year 12 students. Many of the ideas in mirror symmetry are combinatorial in nature, with duality being explored in the context of convex polytopes and/or polygons. These objects are accessible to A-level students, and introducing the idea of duality leads to a certain curiosity in mathematics that can be visually tangible in this case. Seeing the beauty of geometry in this way is often missed in A-levels and could garner interest that can lead to more students taking Mathematics degrees in university. The idea is to create a new interactive workshop for students in the University of Birmingham's repertoire to have a hands-on experience in geometry, execute it in the classrooms first-hand, refine it from this experience, and document it for others to repeat at later dates. This documentation will then be used to perform a training for teachers, hosted at the University of Birmingham to maximise impact.

Finally, I will work with Pride in STEM, a charitable trust for LGBT+ people in Science, Engineering, Tech, and Maths, in the role of public engagement. Pride in STEM hosts public engagement nights titled "Out Thinkers", providing a platform where people can talk about their scientific work while truly being themselves. The aim will be to increase the number of students from underrepresented groups studying mathematics. Moreover, this will aid in helping the public appreciate links between math and physics. Often many people in the audience are young, and such talks can encourage them to take further study or go into academic careers.