Bridging Frameworks via Mirror Symmetry
Lead Research Organisation:
University of Birmingham
Department Name: School of Mathematics
Abstract
Stand in one place. Ask the question "What are the possible ways you could face while standing there?'' One answer is from zero degrees to 360 degrees, but that is not a fully-satisfying answer. The most intuitive answer is you can turn around in a circle. This answer is an example of a geometric classification of possible solutions, or a moduli space. Moduli spaces are ubiquitous in geometry. From conic sections to the range of motion of a robot, one is studying moduli spaces. In algebraic geometry, we study the geometry of the solutions of polynomials and associated geometric classification problems. When one has many variables and uses higher degrees, such questions become difficult. Such shapes formed by Typically there are three ways to study varieties: looking at other objects that sit inside them, finding ways that they sit inside other objects, and finding invariants that help classify them.
In the last 25 years, string theory has giving intuitive frameworks for studying certain classical algebro-geometric objects, Calabi-Yau shapes. In string theory, Calabi-Yau shapes are added to the space-time continuum in order to get physical models for the universe. In mathematics, this led to a geometric duality called mirror symmetry which focuses on the duality between Type IIA and IIB string theory. This rich framework allows many connections between mathematical fields, typically symplectic geometry and algebraic geometry.
Many of the connections made have to do with enumerative geometry, studying how many curves of a certain type sit inside higher dimensional objects. Mirror symmetry turned this problem in symplectic geometry into an algebro-geometric problem, making it easier to compute the answer. Some of the connections sit in number theory. Varieties have number-theoretic analogues where one can study them over a finite field, providing geometric analogues to the Riemann zeta function.
The proposed research plan focuses on finding bridges amongst fields motivated by mirror symmetry. The proposal involves the following projects:
1.) Providing a method to compute the FJRW-invariants in symplectic geometry by linking the invariants to an algebro-geometric setting then using tropical geometry. These invariants describe how many curves of a certain type sit in a generalized version of a Calabi-Yau shape, called a Landau-Ginzburg model.
2.) Studying the number theoretic properties of Calabi-Yau shapes when viewed under mirror symmetry, harnessing properties of the zeta function associated to these shapes.
3.) Classify a certain class of higher-dimensional analogues to polygons by using their correspondence to algebraic objects by using geometric quotients, consequently giving a classification of certain types of Calabi-Yau shapes.
4.) Codify what mirror symmetry means for another type of string theory, heterotic mirror symmetry.
The work presented here will provide more links amongst mathematical fields, creating a more cohesive mathematical community. Each project takes two fields and connects them in a way so that both fields can contribute to the understanding of Calabi-Yau shapes.
In the last 25 years, string theory has giving intuitive frameworks for studying certain classical algebro-geometric objects, Calabi-Yau shapes. In string theory, Calabi-Yau shapes are added to the space-time continuum in order to get physical models for the universe. In mathematics, this led to a geometric duality called mirror symmetry which focuses on the duality between Type IIA and IIB string theory. This rich framework allows many connections between mathematical fields, typically symplectic geometry and algebraic geometry.
Many of the connections made have to do with enumerative geometry, studying how many curves of a certain type sit inside higher dimensional objects. Mirror symmetry turned this problem in symplectic geometry into an algebro-geometric problem, making it easier to compute the answer. Some of the connections sit in number theory. Varieties have number-theoretic analogues where one can study them over a finite field, providing geometric analogues to the Riemann zeta function.
The proposed research plan focuses on finding bridges amongst fields motivated by mirror symmetry. The proposal involves the following projects:
1.) Providing a method to compute the FJRW-invariants in symplectic geometry by linking the invariants to an algebro-geometric setting then using tropical geometry. These invariants describe how many curves of a certain type sit in a generalized version of a Calabi-Yau shape, called a Landau-Ginzburg model.
2.) Studying the number theoretic properties of Calabi-Yau shapes when viewed under mirror symmetry, harnessing properties of the zeta function associated to these shapes.
3.) Classify a certain class of higher-dimensional analogues to polygons by using their correspondence to algebraic objects by using geometric quotients, consequently giving a classification of certain types of Calabi-Yau shapes.
4.) Codify what mirror symmetry means for another type of string theory, heterotic mirror symmetry.
The work presented here will provide more links amongst mathematical fields, creating a more cohesive mathematical community. Each project takes two fields and connects them in a way so that both fields can contribute to the understanding of Calabi-Yau shapes.
Planned Impact
My proposed research will most immediately have an impact on the academic research community, like most research in pure mathematics. That being said, there are many potential long-term applications as the fields being researched have had precedents in being useful in industry and national security. For example, arithmetic geometry and number theory related to Calabi-Yau varieties have been used in cryptography. String theory may bring a grand unified theory of the universe. Tropical methods has helped in mathematical biology and economics. Toric geometry has implications in polytope theory which is used in optimization. More details on these instances are found in the Pathways to Impact statement.
The project has clear directives to disseminate the research and to optimize the possibility that such applications and real-life impact occurs. By giving talks in interdisciplinary conferences both in the UK and elsewhere and by writing more accessible papers, I will achieve a larger readership and a possibility of new real-life consequences. As these projects are intradisciplinary, there tends to be a language shift amongst the disciplines in how they study mathematics and their definitions. I will take extra care in making any article based on this work accessible for all relevant parties alike, fostering a more contiguous mathematical literature.
Work will also be undertaken to foster the next generation of British mathematics through teaching a Part III course on mirror symmetry with an eye towards the style of mirror symmetry done in the UK. Mirror symmetry is a rapidly growing mathematical discipline in which the United Kingdom has positioned itself strategically to be a leader. By teaching the new generation the field, we prepare the next generation of UK mathematicians to continue to lead the world in the field. Also, I will supervise possible research experiences for undergraduates that relate to the combinatorial elements of mirror symmetry to create more independent researchers at the undergraduate level.
As the work in this project can be combinatorial in nature, any effective algorithm to compute valuable invariants that is found will be programmed into Sage and sent to become a Sage package, openly accessible to all interested parties.
The project has clear directives to disseminate the research and to optimize the possibility that such applications and real-life impact occurs. By giving talks in interdisciplinary conferences both in the UK and elsewhere and by writing more accessible papers, I will achieve a larger readership and a possibility of new real-life consequences. As these projects are intradisciplinary, there tends to be a language shift amongst the disciplines in how they study mathematics and their definitions. I will take extra care in making any article based on this work accessible for all relevant parties alike, fostering a more contiguous mathematical literature.
Work will also be undertaken to foster the next generation of British mathematics through teaching a Part III course on mirror symmetry with an eye towards the style of mirror symmetry done in the UK. Mirror symmetry is a rapidly growing mathematical discipline in which the United Kingdom has positioned itself strategically to be a leader. By teaching the new generation the field, we prepare the next generation of UK mathematicians to continue to lead the world in the field. Also, I will supervise possible research experiences for undergraduates that relate to the combinatorial elements of mirror symmetry to create more independent researchers at the undergraduate level.
As the work in this project can be combinatorial in nature, any effective algorithm to compute valuable invariants that is found will be programmed into Sage and sent to become a Sage package, openly accessible to all interested parties.
People |
ORCID iD |
| Tyler Kelly (Principal Investigator / Fellow) |
Publications
Doran C.F.
(2019)
2017 MATRIX Annals
Favero D
(2019)
Derived categories of BHK mirrors
in Advances in Mathematics
Favero D
(2020)
A maximally-graded invertible cubic threefold that does not admit a full exceptional collection of line bundles
in Forum of Mathematics, Sigma
| Description | Project 1 is nearing fruition and dissemination. We have found an interesting development in the study of how one integrates to find FJRW invariants that may allow for a way to apply the Gross-Siebert program to quantum singularity theory. In particular, these integrals (which relate to a IIB string theory) should relate to not just quantum singularity theory, but open quantum singularity theory. The project transitioned into open symplectic geometry and topological recursion instead of tropical geometry. This development, joint with M. Gross (Cambridge) and R. Tessler (Weizmann), has been a great breakthrough for this field. We have results that we are currently writing up ranks one and two, showing this open mirror symmetry and have discovered wall-crossing. These papers are now publicly available on the arxiv at https://arxiv.org/abs/2203.02423 and https://arxiv.org/abs/2203.02435 . This manuscripts are foundational mathematics work that spans over 150 pages. They are the foundation for a new UKRI FLF proposal, won in March 2020. I am now looping back with modern techniques to the original tropical story now with collaborators Silversmith (Warwick) and Cavalieri (Colo State), which may lead to new techniques more in line with the original proposal after this first piece was developed. With regards to project 2, my collaborators and I have found that Arithmetic Mirror Symmetry holds for certain Symmetric Quartic pencils and can find how the motivic viewpoint of periods relates to the point counts on these examples as well as the factors in their zeta functions. This paper is currently being revised for submission. Moreover, we found a more general story: we can realize via string theory that certain polynomials and deformations of polynomials share certain arithmetic properties with regards to solutions of their equations. We can prove this in a preliminary case and it works in any dimension with some technical assumptions. This was published open access in the Israel Journal of Mathematics and accessible on the arxiv at https://arxiv.org/abs/1612.09249 or on the Cambridge repository. We also finished a sequel paper that looks at a more motivic viewpoint. We isolate certain hypergeometric differential equations related to the cohomology of certain varieties over the complex numbers, which we can then use to decompose the L-series when studying the same varieties over finite fields, connecting this viewpoint to more computational number theory. This paper is to appear in the journal Research in the Mathematical Sciences and accessible on the arxiv at https://arxiv.org/abs/1810.06254. Finally, we wrote a book chapter summarising these results for MATRIX 2017. With regards to project 3, methods were found for using the categorization of certain polytopes to effectively classify and categorize all K3 surfaces in toric varieties up to birational and derived equivalence; however, going the other way is difficult. In 2015, I reported the good news that there is an even more interesting viewpoint in the horizon of linking the geometry of objects of differing dimension via derived categories by using polyhedra instead of polytopes. This has been developed and is now on the arxiv. It led to a very general framework that does not need the polyhedral perspective, but is much more easily computed in this context. This work has led to two papers, on in Proceedings of the AMS and another in the open-access journal Algebraic Geometry. Both are on the arxiv and in open access repositories. Lastly, the second part of this proposal EP/N004922/2 and the New Investigator Award again used these techniques to solve a new conjecture (https://arxiv.org/abs/2001.06500). Project four has found new obstructions in the types of deformations studied in the viewpoint of deforming a heterotic string model, which in itself is interesting, but alas adds more risk to the project and more obstructions in the sense of finding a conclusive result. This novel development was found by speaking with string theory colleagues in Oxford on an EPSRC-funded visit. As this was the riskiest objective, and now there is a very complicated cohomological viewpoint to compute, this objective ended up with issues on the technical aspects. Lastly, as a spinoff, a paper was written by Nathan Ilten and myself on Fano schemes of toric complete intersections as a result of my time at the Fields Institute in 2016 (https://arxiv.org/abs/1910.05593), which is submitted. |
| Exploitation Route | Project 1 links Landau-Ginzburg models linked to the current state of the art of open mirror symmetry for Fano manifolds and the main thrust of the Gross-Siebert programme in some sense. It provides new avenues to explore in integrable systems, cluster algebras, open enumerative geometry and mirror symmetry. This is a great result that can be used by many. Project 2 leads to a new method to study the number theory of Calabi-Yau varieties, a generalization of elliptic curves. The work involves many useful tools in number theory, including hypergeometric motives and Dwork cohomology. We bring a physical intuition to the field, hoping to unify number theory and string theory. It is timely, with hypergeometric motives being a timely topic that people are currently exploring. Project 3 can be used to study the derived categories of any complete intersection. It can be used as evidence on whether or not a variety looks like (is birational to) projective space, using Kuznetsov's conjecture for cubic fourfolds as inspiration. We also hope to connect one day with the FanoSearch program, finding an intrinsic understanding of the semi-orthogonal decompositions of the derived categories of all Fano manifolds. |
| Sectors | Other |
| Description | UKRI Future Leaders Fellowship |
| Amount | £1,021,375 (GBP) |
| Funding ID | MR/T01783X/1 |
| Organisation | United Kingdom Research and Innovation |
| Sector | Public |
| Country | United Kingdom |
| Start | 11/2020 |
| End | 10/2024 |
| Description | Podcast invitation with Querdy |
| Form Of Engagement Activity | Engagement focused website, blog or social media channel |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Other audiences |
| Results and Impact | Querdy is a podcast run by Matthew Young, who is a biochemist that recently obtained his PhD. Querdy is his podcast that discusses scientific research and journalism and promotes visibility of LGBTQ+ academics. It is financially supported by the Biochemical Society and the UK charity Pride in STEM. |
| Year(s) Of Engagement Activity | 2019 |
| URL | https://querdypod.com/ep-06 |