Geometry and mathematical physics
The theory of integrable systems studies differential equations which are, in a sense, exactly solvable and possess regular behaviour. Such systems play a fundamental role in mathematical physics providing an approximation to various models of applied interest. Dating back to Newton, Euler and Jacobi, the theory of integrable systems plays nowadays a unifying role in mathematics bringing together algebra, geometry and analysis.
The research of the group includes both classical and quantum integrable systems in relation with representation theory and special functions, as well as algebraic, differential and symplectic geometry.
Academic staff within this group are:
- Dr Hamid Ahmadinezhad: Birational Geometry: Classification of Fano-Mori Spaces in higher dimensions; Applications of algebraic geometry in physics, kinematics and statistics
- Dr Alexey Bolsinov: Integrable tops, bi-Hamiltonian systems and compatible Poisson structures; integrable geodesic flows on Lie groups and homogeneous spaces, magnetic geodesic flows, symmetries and reduction; obstructions to integrability; symplectic and topological invariants for Lagrangian foliations; singularities of the momentum mapping, their invariants and algorithmic classification; projective equivalence in Riemannian geometry.
- Dr Jonathan Eckhardt: Direct and inverse spectral theory of ordinary differential equations, in particular generalised indefinite strings and canonical systems; Inverse spectral/scattering transform method for integrable partial differential equations, long-time asymptotics.
- Professor Jenya Ferapontov: Classical differential geometry (web geometry, projective differential geometry, Lie sphere geometry, theory of congruences, conformal structures, geometric aspects of PDEs); Integrable systems (equations of hydrodynamic type, hyperbolic systems of conservation laws, multi-dimensional dispersionless integrable systems, Hamiltonian formalism, symmetry methods).
- Professor Sara Lombardo: Integrable systems, Automorphic Lie Algebras, Analysis of Integrable PDEs, Nonlinear Waves.
- Dr Vladimir Novikov: Classification of integrable nonlinear partial differential equations and differential-difference equations. Integrability tests: symmetry approach, perturbative symmetry approach in the symbolic representation. Integrable models of mathematical physics.
- Dr Artie Prendergast-Smith: Algebraic geometry, especially birational geometry, classification of varieties, minimal model theory.
- Dr Elisa Postinghel: Algebraic Geometry and in particular Birational Geometry: positivity, Newton-Okounkov bodies, the minimal model program, linear systems on higher dimensional varieties. Secant varieties and tensor decomposition. Tropical geometry.
- Dr Alan Thompson: Algebraic geometry, especially the geometry of surfaces and threefolds: moduli of K3 surfaces and Calabi-Yau threefolds, construction of Calabi-Yau threefolds, fibration structures, mirror symmetry, and the minimal model program.
- Professor Sasha Veselov: Classical and quantum integrable systems in relation with geometry and representation theory; solvable Schroedinger equations, special functions and Huygens' principle.