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UK funding (£595,178): Open Mirror Geometry for Landau-Ginzburg Models, Renewal Ukri1 Oct 2025 UK Research and Innovation, United Kingdom
Overview
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Open Mirror Geometry for Landau-Ginzburg Models, Renewal
| Abstract | Mirror symmetry is a modern research discipline that aims to mathematically prove a duality from string theory establishes new theorems in geometry. In particular, mirror symmetry links two areas of geometry by predicting that the symplectic geometry of a given space M is encoded in the algebraic geometry of a so-called mirror space M*. It has inspired deep theorems in mathematics and solved centuries-old problems in enumerative geometry. Moreover, it provided a pathway towards establishing new foundations which encode the study of geometric disciplines like symplectic topology and algebraic geometry using the language of homological algebra, following the ideas of Fields Medallist Kontsevich. Mirror symmetry transcends mathematical disciplines, having used techniques from geometry, algebra, combinatorics, integrable systems, number theory and mathematical physics. Given a symplectic space M, the first question we have to answer is how to construct a conjectural mirror space M*. Afterwards, one aims to prove that the mirror phenomenon occurs, i.e., the study of symplectic geometry of M are encoded by the algebro-geometric study of M*. Historically these two questions are handled separately, but the modern approach to constructing mirrors aims to build the mirror directly from geometric data of M, handling both steps in realising mirror symmetry simultaneously. Broadly speaking, this fellowship develops foundations towards building the mirror intrinsically in the case where M is a Landau-Ginzburg model. Roughly, a Landau-Ginzburg (LG) model is a function encapsulating geometry in its singularity theory. In the last decade, they have become crucial to the understanding of mirror symmetry as they can be found naturally by deforming certain spaces in symplectic geometry and algebraic geometry. However, they are interesting in their own right in the field of non-commutative algebraic geometry. In the first part of the fellowship, the open (and closed) enumerative geometry of Landau-Ginzburg models have been explored. Jointly with Ran Tessler and Mark Gross, we established a new approach to construct mirrors for (certain) Landau-Ginzburg models. This is done by building / generalising an open enumerative theory for Landau-Ginzburg models and writing a mirror LG model using the open enumerative invariants computed. The primary outputs of the first part of the fellowship included: developing an open enumerative theory in dimension two for Fermat polynomials, proving the first open mirror symmetry theorem for LG models, establishing the first wall-crossing structures for enumerative geometries for LG models, proving a new type of open topological recursion relation, and establishing a formula for primary genus-zero r-spin invariants. This renewal continues this investigation, breaking past dimension two. This involves new rich mathematical structures and develops closer tether to the approach to mirror symmetry via homological algebra by Kontsevich mentioned above. |
| Category | Fellowship |
| Reference | MR/Y033841/1 |
| Status | Active |
| Funded period start | 01/10/2025 |
| Funded period end | 30/09/2028 |
| Funded value | £595,178.00 |
| Source | https://gtr.ukri.org/projects?ref=MR%2FY033841%2F1 |
Participating Organisations
| Queen Mary University of London |
The filing refers to a past date, and does not necessarily reflect the current state. The current state is available on the following page: Queen Mary University of London, London.
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