Mathematicians have built a new mathematical mirror that lets them see hidden geometric structures in spaces too complex to study directly. This research tackles a core problem in pure mathematics: how to prove that two seemingly different geometric worlds—symplectic geometry (the study of spaces that preserve area) and algebraic geometry (the study of shapes defined by equations)—are actually two sides of the same coin. The idea, called mirror symmetry, emerged from string theory and has already solved problems that stumped mathematicians for centuries. But constructing the "mirror" of a given space has been a two-step process: first guess the mirror, then prove it works. This fellowship develops a method to build the mirror directly from the original space's geometry, handling both steps at once. The work is fundamental science with no immediate practical application. It focuses on a class of objects called Landau-Ginzburg models—functions whose singularities encode rich geometric information. The researcher has already proven the first open mirror symmetry theorem for these models in two dimensions. This renewal extends the approach beyond dimension two, forging closer links to the homological algebra framework pioneered by Fields Medallist Kontsevich. Past fundamental work in mirror symmetry has unexpectedly influenced fields from cryptography to quantum computing; deeper understanding of these geometric dualities could eventually reshape how mathematicians and physicists think about the structure of space itself.
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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.
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