Geometric methods serve as powerful tools for explaining the behaviour of physical systems, both classical and quantum. For example, in a program of research that we like to call "geometric quantum mechanics'', methods of algebraic geometry, symplectic geometry and complex manifold theory are used to provide a novel basis for the description and explanation of quantum phenomena. Physical characteristics of quantum systems can be represented by various geometrical objects that are preferentially identified in the manifold of pure quantum states or in the tangent bundle of that space. This allows one to gain a deeper understanding of quantum mechanical notions such as entanglement, measurement, and uncertainty. From a mathematical perspective one can view quantum theory as an exploration of (a) the algebraic geometry of complex projective space when this space has been enriched with the Kählerian geometry of the Fubini-Study metric, and (b) the profound relation of this geometry to various aspects of modern probability theory. According to this view the various algebraic sub-varieties of complex projective space, along with various other geometrical entities, admit a systematic interpretation as elements of the physical theory.
New insights can be gained concerning the measurement problem, quantum information, quantum control, constrained quantum systems, and quantum statistical mechanics. In particular, the geometrical approach offers surprising insights concerning (a) generalised measurements, including so-called SIC-POVMs, and (b) quantum systems in thermal equilibrium. The geometrical approach also acts as a natural point of departure for other topics currently under investigation such as (a) foundational issues in quantum theory, and (b) the development of physical theories that extend quantum mechanics in various ways. The latter include, for example: stochastic models for the collapse of the wave function; nonlinear models of the Kibble-Weinberg type; and novel approaches toward a better understanding of the interrelationship of the geometry of quantum theory and the geometry of space-time, building on and generalising various well-established constructions coming from Penrose's twistor theory.
Geometrical methods can help one understand common traits in mechanical systems that seem at first sight very different from one another. A celebrated example was discovered in 1966 by V. I. Arnold: both the motion of a rigid body and the evolution of an ideal incompressible fluid can be described by geodesics (length-minimizing curves) on certain spaces. While the mathematical details vary between the two systems, the geometric nature of the evolution—geodesic motion—is shared by both. If a mechanical system exhibits symmetries, one can eliminate the corresponding degrees of freedom. The question of how exactly this can be done has led to the mathematical theory of symmetry reduction. One of the topics in this active area of research currently being pursued at Brunel is concerned with mechanical systems whose variational descriptions depend on higher-order derivatives of the evolution curve. A number of first-order symmetry reduction methods have been extended to these systems, such as the so-called Euler-Poincaré reduction for systems on Lie groups. For example, an interesting problem in quantum control has been studied using these methods: a quantum mechanical system is steered along a smooth trajectory through a number of target states, in such a way that the Hamiltonian satisfies a “least change” constraint. Geometrically, this can be interpreted as an optimal curve-fitting problem on complex projective space.