ObjectivesThe main objective of the project is to open a new direction of research of geometrical analysis in Lie groups using the following instruments: * geometric measure theory in Lie groups; * differential equations in Lie groups and Cauchy Riemann (CR) manifolds; * geometric numerical integration and Lie groups.Results from these fields will have a strong impact on emergent technologies, in particular in modelling: * magnetic resonance tomography and quantum computation; * mathematics of hearing; * neuro-mathematics and visual perceptions.Geometrical analysis in Lie groupsSub-Riemannian geometric analysis in Lie groups models the motion of a system in which some directions are not allowed by a constraint, that is not necessarily a physical constraint but that can be a differential one.The first goal is then is to understand the differential geometry of the relevant objects in Lie groups with a subriemannian geometry such as curves, surfaces and manifolds and define their properties. The differential calculus used here is particularly applicable to the description of the curved surfaces found on manifolds.The second goal concerns geometric measure theory, which studies the stability of the objects on the manifolds.Once the basic objects in the Lie groups are understood, the next step is to develop the partial differential equations (PDEs) to describe their motion. It is the integration of differential geometry, geometric measure theory, and PDEs in Lie groups that will open up a wide range of applications for solving real problems.Applications to emergent technologiesRecent advances in neurophysiology treat the brain as a highly anisotropic structure where cortical neurons are modelled as particles, like electrons and photons.GALA will model the electric charge flow from sensory neurons in the visual cortex and relate this flow to the geometric structures of perceptive representations thus contributing to understanding of how vision occurs.Similarly, modelling the signal-processing functions of the cochlea the sensory organ in human ears will contribute to the mathematics of hearing and may lead to new hearing aids.In materials science, impedance tomography is used to determine the bulk properties of a body through surface electrical measurements a promising non-destructive technique with numerous potential industrial applications.Many material composites exhibit complex micro-geometries and the GALA instruments will help understand these offering the prospect of characterising the macroscopic behaviour of materials using minimal information.
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|Titolo:||Geometric Analysis on Lie groups and Applications|
|Autori:||Giovanna Citti; Sergio Polidoro|
|Data di pubblicazione:||2006|
|Appare nelle tipologie:||Partecipazione a progetti di ricerca|
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