# Research

Research in the AGIS group spans a wide range activities across pure mathematics, applied mathematics and mathematical physics, with particular emphasis on the links between these disciplines. The AGIS group maintains a strong interest in pure research problems in its constituent fields (for example: homological algebra, hyperkaehler geometry, inverse scattering theory), but actively seeks to develop in the interfaces between fields (for example: algebraic geometry of Calogero-Moser spaces, applications of differential geometry to soliton dynamics, applications of representation theory to statistical mechanics). This makes the group a particularly vibrant and exciting place to study for a research degree. Our main research strengths are:

#### Algebra and representation theory

*Chalykh, Crawley-Boevey, Faber, Marsh, Martin, Parker, Tange*

- Homological algebra, tilting theory, derived categories, triangulated categories
- Kac-Moody Lie algebras, quantum groups, Coxeter groups, cluster algebras
- Noncommutative algebra – Cherednik algebras, diagram algebras, group algebras, Hecke algebras, quiver algebras, etc. – and links with algebraic geometry and representation theory
- Representations and invariants of algebraic groups, algebraic group actions
- Representation theory of finite-dimensional associative algebras and quivers

#### Low dimensional geometric topology

*Faria Martins, Marsh, Martin, Parker
*

- Braid groups
- Diagram algebras and categories
- Dimer models and surface cluster algebras
- Topological phases of matter and topological quantum computing
- Topological quantum field theory

#### Connections of logic to algebra and geometry

*Brooke-Taylor, Gambino, Macpherson, Mantova*

Mathematical logic links to algebra and geometry in many ways, but especially through model theory and category theory. Model theory concerns the study of mathematical structures (e.g. groups, rings and fields) from the viewpoint of first order logic (first order expressibility of properties, the combinatorics and geometry of `definable sets’). Category theory studies mathematical objects by focusing mappings between them. This point of view led to the formulation of general concepts (adjunctions, limits, Kan extensions) that allow us to link different areas of mathematics (including algebra, geometry, topology, logic) and find applications also in theoretical computer science.

- Permutation groups, automorphism groups, model theory of group and fields
- Connections of model theory to number theory
- Surreal numbers, ordered fields with additional structure
- Categorical and homotopical algebra, operads, higher-dimensional categories.
- Connections to homotopy type theory via categorical logic.

#### Geometric variational problems

*Harland, Kokarev, Speight, Wood*

Variational problems are ubiquitous in differential geometry and mathematical physics, where the “best” or “most natural” objects often minimize energy, in some sense. This work concerns the existence, stability, construction and geometric properties of critical points of geometrically natural energy functionals in a variety of contexts.

- Harmonic maps and morphisms between Riemannian manifolds, and their generalisations
- Gauge theory and Yang-Mills-Higgs theory
- Geometry of moduli spaces of critical points; hyperkähler geometry
- Moduli spaces of topological solitons
- Minimal surfaces and extremal eigenvalue problems

#### Integrable systems theory

*Buryak, Caudrelier, Chalykh, Fordy, Mikhailov, Nijhoff, Ruijsenaars*

Integrable systems theory is concerned with systems of PDEs and ODEs which can, in some sense, be solved exactly. Integrable systems have a rich and fascinating mathematical structure making them worthwhile objects of study in their own right, quite apart from their innumerable links with geometry and algebra, where they have motivated many important and influential developments (e.g. twistor theory, quantum groups). Topics of interest at Leeds include:

- Lax pairs, inverse scattering transforms, Darboux and Bäcklund transforms, soliton theory
- Symmetries and algebraic theory of differential equations
- Special function theory and Painlevé equations
- Hamiltonian systems and Poisson algebras
- Integrable many body problems, Calogero-Moser spaces
- Quantum integrable systems and quantum algebras
- Applications to the geometry of moduli spaces of curves
- Integrable variational problems and Lagrangian multiform theory

#### Discrete systems and difference equations

*Caudrelier, Chalykh, Fordy, Nijhoff, Ruijsenaars*

Discrete systems arise in various branches of physics, where they model coarse crystalline structures, and as analogues of continuum systems of PDEs and ODEs. In both contexts, one can argue that the discrete system is more fundamental than its continuum limit. Their study introduces new challenges, and has inspired new developments in other areas of mathematics, such as discrete complex function theory and discrete differential geometry. Our work in this area concerns:

- Integrability of high-dimensional discrete systems, discrete differential geometry
- The Laurent phenomenon and connexions with quiver mutation
- Birational maps
- Spatially discrete solitons
- Integrable PDEs on graphs

#### Discrete and spectral geometry

*Kokarev, Houston, Strohmaier*

Spectral geometry lies at the intersection of differential geometry, partial differential equations, and analysis. It finds applications in other areas of pure mathematics such as number theory and the theory of dynamical systems, and is to a large extent motivated by questions originating in the study of real-life phenomena, such as vibration, heat propagation, oscillation of fluids, and quantum mechanical effects. For example a model question, posed by Mark Kac half a century ago, is “Can one hear the shape of a drum?” The challenge is to understand the geometry of an idealised drum from the set of its vibration frequencies, that is the Laplace eigenvalues. Kac’s question is closely linked to other topics in spectral geometry, including shape optimisation, spectral asymptotics, and spectral invariants.

The concepts and methods of spectral geometry are amenable to the setting of the discrete geometry and extremely important for applications. For example, bounds for the Laplace-Beltrami eigenvalues and various properties of its eigenfunctions are used in image processing, shape recognition, and machine learning. Currently, Kevin Houston is using the discrete Laplace-Beltrami operator in the study of biological shapes, for example skull and jaw shapes.