Open Mirror Geometry for Landau-Ginzburg Models

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.

In the Bond Report published in 2018, facilitated by the EPSRC, the author outlines the value of pure mathematics in its own right. This proposal, at its onset, provides new transformative results in mirror symmetry and geometric fundamental research. Pure and fundamental mathematical research often takes a long time to directly impact society; however, certain actions can be done in the intermittent time to facilitate knowledge exchange and expedite these timescales. This project has a strong dissemination strategy outlined in the "Pathways to Impact" that will foster a healthy mathematical dialogue in the advent of potential applications.

The most obvious impact for this research is that it will retain and grow world-leading research activity. By putting the UK at the forefront of mathematical research, we gain prestige in the global market of mathematicians, allowing us to attract the internationally leading faculty and students at both the undergraduate and graduate level into the UK. This develops the next generation, and many of these students will go on into UK academia and industry. This proposal will in turn recruit two high-tier postdoctoral research associates to the UK to create original research outputs in a field of UK strength. The workshops that are a part of this grant work as both dissemination of research related to this fellowship but also key recruitment and retention tools for research students to consider academic positions in the UK and in the Midlands.

I have a passion for increasing inclusion amongst underrepresented groups in mathematics. This started in 2008 as a math team coach at a 78% BME high school in the South in the USA. This continued when I was a founding graduate coordinator in 2013 of the Emerging Scholars Program at the University of Pennsylvania, a program to recruit underrepresented undergraduate students into STEM fields targeting specifically women. I built peer-led interactive workshops for students to learn more about higher-level mathematics. Now, I am the local organisation lead for the 2020 LGBT STEMinar conference at the University of Birmingham, a one-day conference to promote visibility of UK LGBTQ-identified researchers in STEM. This conference lately has had attendance around 150 each year. I am also the staff advisor for the only currently running oSTEM (out in STEM) undergraduate organisation in the UK. This gives momentum for me to do LGBTQ outreach and public engagement in mathematics, an avenue that has not been significantly explored in the UK by mathematical learned societies, despite the BBC Scientist of the 20th century (Alan Turing) being a gay mathematician.

The outreach and public engagement in this fellowship proposal has two approaches. One is through performing public engagement events through Pride in STEM and their speakers' nights "Out Thinkers." I will create a new Out Thinkers venue in Birmingham's gay village to tap into a community that typically does not get targeted engagement with science.

Secondly, I will take advantage of the opportunity the FLF provides to do a more transformative engagement project. I will initiate a multidisciplinary project of creating contemporary art using ideas from my results in quantum mathematics, geometry, and number theory with artist Josh Westerman. Westerman has an Masters degree in Experimental Sound Practices and Integrated Media. We will create installation art that represents and explains my results on a more intuitive and emotional level, bypassing heavy mathematics at first and then allowing audiences to choose the level of mathematical depth they want to explore. We aim to exhibit art in LGBTQ venues as well as in standard art spaces as specified in "Pathways to Impact."