GEOMETRIC FLUID DYNAMICS
Continuum fluid models are highly relevant in many areas of physics and technology, including turbulence, meteorology, astrophysical plasmas and nanoparticle dynamics (among many others). In particular, the topics covered within this direction are listed below.
Geometry of fluid models: geometric aspects of fluid systems are considered. 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 research focuses on applications of these concepts, particularly in image processing and shallow water waves.
Hybrid kinetic-fluid models: these models couple conventional fluid models with kinetic equations governing either a gas coexisting with the fluid (e.g. in the context of magnetic confinement fusion) or fluctuation dynamics (e.g. in the context of turbulence). Various aspects of these models are studied, including their symmetry properties and formal stability considerations.
Structure-preserving integrators: various geometric numerical schemes will be discussed, with special emphasis on variational integrators. Their applications in fluid models are of central interest to the network and possibilities of their applications to hybrid kinetic-fluid models are also studied. In addition, structure-preserving numerical methods are considered, especially in terms of discrete and finite element exterior calculus.
SYMMETRY AND SHAPE ANALYSIS
Rapidly developing medical-imaging technologies are revealing exquisitely detailed descriptions of human anatomy in a wide variety of data structures, often at several levels of resolution. Computational Anatomy (CA) offers a unified approach using smooth invertible transformations specific to each type of data structure. In particular, the topics covered within this direction are listed below.
Data structure and fusion: unified approaches for registering images encoded in a wide variety of data structures. The unifying concept is the momentum map from Lie group theory. The research explores the question of how to synthesize (fuse) information by accounting for the different transformation properties of the data structures.
Multiple resolutions and dynamic images: the momentum map framework will be considered for enabling registration of data at multiple resolutions by concatenating Lie group transformations. Time-varying images will be treated in this geometric framework by matching snapshots in time using geodesic splines that interpolate the image snapshot from one time to another.
Changes in image topology: Extensions of the transformative approach to allow changes in topology in the course of passing from one image to another by using the method of metamorphosis, an innovative reduction technique that applies to coupled variational systems.
QUANTUM DYNAMICS AND SYMMETRY
The use of reduction by symmetry in geometric quantum dynamics stands as one of the main novelties of this network, which also encompasses applications in geometric quantum control. In particular, the topics covered within this direction are listed below.
Reduction by symmetry: applications of both Euler-Poincaré (EP) and Lagrange-Poincaré (LP) to the fundamental descriptions of quantum mechanics (Schrödinger, Heisenberg and Dirac pictures). Both EP and LP approaches shed new light on the geometric interpretation of the geometric phase and its non-Abelian counterparts. Moreover, applications in quantum control will be studied, along with other quantum applications of geometric flows (e.g. Fubini-Study geodesics; Ricci and Renormalization group flows in two-dimensional sigma-model theories).
Quantization and quantum groups: the role of symmetry will be discussed within quantization procedures (e.g. Stratonovich-Weyl quantization) and quantum gravity aspects (e.g. the so-called consistent truncations by symmetry). Also, special attention will be addressed to applicability of algebraic techniques common to quantum mechanical systems with extended symmetry. Recently emerged connections between Hopf algebras, braid groups and quantum computation are also considered.
Coherent states and momentum maps: new expressions of momentum maps will be discussed in relations to quantum coherent states (particularly, standard and squeezed coherent states). This provides a new contribution to the geometric interpretation of coherent states, which will also be discussed in the context of non-Hermitian quantum mechanics.