# A UK Research Network

Geometric mechanics arises mainly from the observation that most of the natural laws are governed by symmetry principles. This applies to both Hamiltonian/Lagrangian systems and certain types of dissipative systems. Over the years, all these systems were shown to allow for a unifying framework within the context of geometric mechanics. This, in turn, led to the development of a number of different directions, including geometric control theory, geometric fluid dynamics, variational integrators, and so on.

In recent years, yet another promising direction has emerged, known as *Geometric Quantum Mechanics*. Geometric mechanics has lead to insights into the interaction of quantum and classical systems, and points towards a hybrid description of such systems. The use of geometrical methods in quantum systems has also found applications to control of quantum systems, measures of entanglement, and parametrisation of generalised coherent states, to name but a few.

The study of *Stochastic Geometric Mechanics* aims to extend geometric mechanics of classical (deterministic) dynamical systems to the case of systems for which random phenomena must be taken into account. By appropriately coupling the stochastic noise to the classical system, one can obtain stochastic analogues of the conservation laws appearing in classical geometric mechanics, and using these describe the evolution of the system in terms of lower-dimensional “reduced” spaces. Already there is a clear description of stochastic Euler-Poincaré reduction (which applies to such disparate physical systems as rigid bodies, fluids, and plasmas), and there has been progress on extending the stochastic treatment to other classes of systems (such as systems with non-holonomic constraints). Stochastic Geometric Mechanics has also been used for both uncertainty quantification and data assimilation for fluid mechanics and geophysical models. However, there remains much to be done in this area.

The topic *Symmetry and Physics* applies more broadly to the study of geometrical and symmetry methods in physics. These include the study of coherent states in semiclassical descriptions of quantum systems, systems on noncommutative spacetime, the use of non-Hermitian techniques in quantum systems, aspects of algebraic geometry in application to, and interacting with, gauge theory as well as string theory, such as Calabi-Yau manifolds, holographic correspondences and string phenomenology, entanglement in many-body systems and its connection with geometric quantum mechanics, and the study of quantum integrable systems.

The AGM Network brings together two leading groups in applied geometric mechanics in the UK, respectively Imperial College and the University of Surrey (where the AGM network is coordinated). This year we have the good fortune of being joined by colleagues at City, University of London, and we are looking forward to the fresh perspectives they will bring to the joint meetings. It is hoped that joining these task forces will impact the future of geometric mechanics in the UK and abroad.

* First Meeting*: Geometric Quantum Mechanics

*--*University of Surrey, 25 April 2019

* Second Meeting*: Stochastic Geometric Mechanics: fluid models and uncertainty quantification -- Imperial College London, 2 December 2019

* Third Meeting*: Symmetries in Physics -- City, University of London, 11 December 2019