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Image: D.D. Holm, T. Schmah & C. Stoica. Geometric Mechanics and Symmetry:

From Finite to Infinite Dimensions, Oxford University Press (2009).

### Second AGM Meeting

Geometric Hydrodynamics: Quantum and Classical

The focus on hydrodynamic models finds its justification in the high importance that continuum fluid models possess within many areas of physics and technology, including turbulence, meteorology, plasmas, and quantum systems (among many others). In particular, an innovative direction will be pursued in terms of hybrid quantum-classical models. The latter represents a long-sought method in molecular dynamics: as the curse of dimensions in many-particle quantum systems poses severe computational challenges, some degrees of freedom are treated as classical while other are left quantum. This leads to the unanswered question of quantum-classical coupling. Other directions will also involve nonlinear waves, oceanography, and modulation theory. The apparently diverse set of topics in this endeavour all share different concepts in geometry and symmetry that serve to unify the derivations of their fundamental equations and the interpretations of their solutions. For example, hydrodynamic models and methods of data assimilation for weather prediction are also used in fundamental applications of shape analysis for the comparison of biomedical images. Likewise, the study of geodesic flows in geometric mechanics (from rigid body to fluid dynamics) is of central importance also in the analysis of the Fubini-Study geodesics in quantum dynamics.

Topics expected to be discussed include:

(1) Geometry of fluid models: Geometric aspects of fluid systems will be discussed. These involve highly mathematical concepts in Poisson geometry, such as dual pairs, as well as geometric flows such as geodesics on diffeomorphism groups and vortex filament dynamics. Part of the meeting will discuss applications of these concepts, particularly in multiscale systems and shallow water waves. Stability issues will also be considered along with the theory of Relative Equilibria. Euler-Poincaré variational principles for the generalised Green-Naghdi and Whitham equations will be discussed.

(2) Hybrid quantum fluid models: These models couple fluid models describing a classical subsystem with quantum equations governing the degrees of freedom of a quantum component such as electrons or spins. Various aspects of these models will be discussed including their geometric properties and formal stability considerations.

The date of the meeting, timetable, participants, speakers, and further details will be posted here.

The London Mathematical Society administers a Caring Supplementary Grant Scheme. Further information about this scheme can be found on the LMS website: www.lms.ac.uk/grants/caring-supplementary-grants

For any enquiries, please contact Hamid Alemi Ardakani.