Mathematicians in Leeds are building a unified framework to connect algebra, geometry, and topology—three branches of mathematics that have historically developed in isolation. The problem is that powerful geometric and combinatorial patterns, first noticed in representation theory (the study of how abstract algebraic objects can be expressed in more concrete terms), have recently cropped up in fields as diverse as mathematical physics and low-dimensional topology. No one yet understands why these patterns appear across such different areas, or how to systematically exploit them. The team aims to explain these connections and place them within a single, coherent theory. This is fundamental science with no immediate practical application. But similar work in the past—for example, the development of group theory or the mathematics behind error-correcting codes—eventually underpinned everything from cryptography to satellite communications. A deeper understanding of these algebraic-geometric-topological interfaces could, in the long term, provide the mathematical tools needed for advances in topological physics, materials science, or engineering.
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A fundamental, and often successful, way of studying an abstract mathematical object is to consider methods of representing it in another, more concrete object. This is a powerful idea, and recent progress in algebraic representation theory and related areas has given rise to strong opportunities for the transformation of other fields. In particular, geometric and combinatorial phenomena initially specific to representation theory have emerged in many other fields, leading to effective new techniques and applications. Our team is at the forefront of these developments. The PI and the five CoIs have contributed to major advances in the past decade, with their expertise ranging from algebra, geometry, and topology to mathematical physics. This provides new ways to link algebra and geometry & topology. Examples include the categorification of the Grassmannian cluster structure, the McKay correspondence for reflection groups, the lifting of Lie-theoretic techniques to 2-dimensional category theory, with applications to topological physics, and the derivation of decomposition matrices of Brauer algebras from generalised Lie geometry. In all cases, the medium for interpolating between the theories is an emergent geometrical property which is not well understood. For the advancement of research, there is a strong need for explaining these phenomena and placing them in an encompassing novel paradigm. Our proposal hence seeks to understand and investigate relations between very different areas, and so to push on from there in a more systematic framework. This aim would benefit from a broad, holistic view of representation theory, embracing Lie theory, algebraic geometry, low dimensional topology and mathematical physics. Our team in Leeds is uniquely qualified to pursue this programme. Together with specialist collaboration of many mathematicians at our international partner institutions, we will address the current challenges, provide solutions to open questions and develop applications by establishing bridging to other fields. We are in a position to embrace the perspectives of both pure and application-driven mathematics, and with the potential, in the long term, for serving the needs of physical sciences, life sciences and engineering. This unification of perspectives requires a programme-level research structure and algebra is the right core platform for such an ambitious venture. Thus our proposal will push forward the mathematical state-of-the-art and will shape the future directions in the areas we touch upon.
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