Mathematicians are building a new kind of counting system for abstract spaces, using vector spaces instead of plain numbers to reveal hidden symmetries in geometry and physics. Traditional invariants—like counting the holes in a coffee cup—can tell two objects apart but often miss deeper relationships. This project develops "cohomological BPS invariants," which take values as vector spaces rather than single numbers. Because vector spaces have non-negative dimensions, these invariants have already proven central positivity conjectures in algebra and combinatorics. The goal now is to go further: by adding algebraic structure to these vector spaces, researchers expect to uncover symmetries between different invariants and prove long-standing conjectures in algebraic geometry. This is fundamental, curiosity-driven mathematics with no immediate practical application. However, the work directly addresses predictions from string theory about the number of algebraic curves inside three-dimensional manifolds called 3-Calabi-Yau varieties—objects that have driven thirty years of geometric theory. Success would also improve understanding of quantum scattering diagrams, which are central to cluster algebras and mirror symmetry. Past fundamental work in this area has unexpectedly reshaped fields from cryptography to particle physics, and deeper algebraic foundations here could eventually enable similar cross-disciplinary breakthroughs.
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Enumerative invariants appear throughout mathematics and mathematical physics. They give a concrete way for us to tell objects apart: if two objects have different enumerative invariants, they are fundamentally distinct. For example, the number of holes in a space is an enumerative invariant from topology that enables us to distinguish between a coffee cup and a glass. In physics, BPS invariants are invariants of a space that determine the numbers of particles of different kinds predicted by the physical theory built from that space. My research programme involves a mathematical formulation of these invariants called cohomological BPS invariants, which have found applications across a broad swathe of algebra, geometry and topology. Unlike traditional invariants that take numerical values, the values of cohomological are vector spaces; we can recover the enumerative invariants by computing dimensions of these vector spaces. Reflecting the fact that the dimension of a vector space is never negative, cohomological BPS invariants have been used in the last decade to prove central positivity conjectures in algebra and combinatorics. Furthermore, endowing these vector spaces with extra algebraic structure is expected to produce isomorphisms between different cohomological BPS invariants and thus equalities at the level of enumerative invariants, or symmetries between them. Similarly, upgrading cohomological BPS invariants themselves to representations of basic Lie algebras should enable us to deduce very general results about the symmetries of enumerative invariants from the representation theory of those Lie algebras. The goal of this research programme is to fulfill this promise, and develop and exploit approaches to BPS cohomology that enable us to see beyond positivity results to a theory that explains these extra algebraic structures and extra symmetries in enumerative geometry and beyond. The foundational objective is to develop a new, more powerful construction of BPS cohomology and BPS Lie algebras, using a combination of quantum algebra and geometric representation theory. This is a branch of algebra that is inspired by the study of quantum field theory, and which builds algebraic operations out of geometric ones. The research programme will have multiple applications to algebraic geometry. For over thirty years, geometers have been developing theories of enumerative invariants in order to explain and prove startling predictions by string theorists regarding the counts of algebraic curves inside certain three-dimensional complex manifolds called 3-Calabi-Yau varieties. In order to translate the physical theory into algebraic geometry, several extremely rich theories associated to these manifolds have been developed. Many of the deepest conjectures in the subject posit equivalences between these mathematical theories, and symmetries within them. A key application of this project will be to prove these conjectures using new foundations for BPS cohomology. The programme will also develop applications to understanding quantum scattering diagrams. As well as being a fundamental tool in calculating scattering amplitudes in string theory, such diagrams are a central tool in cluster algebras and homological mirror symmetry, bridging algebraic and symplectic geometry. By establishing enhanced foundations for the definition and study of cohomological BPS invariants this research programme will transform our ability to use ideas from physics, as well as powerful algebraic constructions, to uncover hidden symmetries of enumerative invariants and prove conjectures in a range of applications in algebraic geometry, representation theory, and beyond.
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