# School of Mathematics

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- School of Mathematics
- Research
- Groups
- Algebra, Geometry and Integrable Systems

## Algebra, Geometry and Integrable Systems (AGIS)

In modern mathematics and mathematical physics the areas of algebra, geometry and integrable systems have become inextricably interwoven. The School of Mathematics is in a unique position to exploit this overlap as it harbours expertise in all three areas, and has a lively tradition of exchange and collaboration between them. This tradition was recently formalized with the formation of a single research group, AGIS, with a common colloquium series.

### In this section

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:

**Noncommutative algebra and representation theory**Crawley-Boevey, Marsh, Martin, Parker, Tange

- Group Theory
- Homological algebra, tilting theory, derived categories, triangulated categories
- Kac-Moody Lie algebras, quantum groups, cluster algebras
- Noncommutative algebra - Hecke algebras, group 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

**Geometric variational problems**Harland, 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 symplectic analogues
- Gauge theory and Yang-Mills-Higgs theory
- Geometry of moduli spaces of critical points. Hyperkaehler geometry.
- Moduli spaces of topological solitons

**Integrable systems theory**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 Backlund transforms, soliton theory
- Symmetries and algebraic theory of differential equations
- Integrable many body problems, Calogero-Moser spaces
- Quantum integrable systems and quantum algebras

**Discrete systems and difference equations**Chalykh, Fordy, Nijhoff, Ruijsenaars, Speight

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 been the focus of intense activity in recent years. 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

**Discrete differential geometry**Houston

Differential geometry is concerned with the differential properties of manifolds, in particular surfaces. In the real world one usually does not have a manifold but a collection of points that lie on a supposed manifold (say from a scanned image). One can make these points into a triangulation. The question is, since differential geometry is so successful at describing surfaces, can we carry over notions from it to triangulations, i.e., the discrete setting?

One fruitful area of study has been in generalizing the Laplace-Beltrami operator. In the differential setting this allows one to say a lot about a manifold. In the discrete case it allows us to analyse, smooth, compare and match shapes. Currently, Houston is using this operator in the study of skull shape.

For further information please see our Group pages.

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