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Geometrical Analysis in Lie groups and Applications

Objective

The GALA is an Adventure STREP that will pioneer explorative instruments of mathematical geometric analysis in Lie groups and will provide on solid analytic basis novel modellistic tools for applications to vision and hearing and to magnetic resonance tomography. Sub Riemannian geometric analysis in Lie groups is an innovative field of scientific research, which considers the description of strongly non- isotropic systems.

In particular it 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 e.g. a magnetic field. The allowed directions of motion are called horizontal directions and are described through vector fields. These vector fields play t he role of derivatives in classical calculus all the theorems of analysis and differential geometry have to be rephrased in terms of these new instruments. We will need to define from a purely mathematical point of view, the main properties of the objects of the space, with instruments of Sub-Riemannian differential geometry, anisotropic partial differential equations of sub-elliptic and ultra-parabolic type and geometric measure theory in Lie groups. Results in this field will allow facing long- standing o pen problems in mathematics, which cannot be dealt with standard instruments. We will see that these instruments can be used to formalize models of the functional geometry of the cortex, and in particular to describe vision and hearing.

Magnetic MRI induces rotation in the spin of the atoms and will be described in the Lie group of rotations. Quasi-linear mapping will describe composite media in material science. Finally we will need to develop numerical instruments in this anysotropic setting, in order to validate our results. The interaction within an interdisciplinary team will help in individuate the geometrical and analytic aspects more suited for the description of the different problems.

Call for proposal

FP6-2004-NEST-C-1
See other projects for this call

Funding Scheme

STREP - Specific Targeted Research Project

Coordinator

ALMA MATER STUDIORUM - UNIVERSITÀ DI BOLOGNA
Address
Via Zamboni 33
Bologna
Italy

Participants (7)

HELSINGIN YLIOPISTO
Finland
Address
Yliopistonkatu 4
Helsingin Yliopisto, Helsinki
UNIVERSITAET SIEGEN
Germany
Address
Walter-flex-strasse 3
Siegen
NORGES TEKNISK NATURVITENSKAPELIGE UNIVERSITET
Norway
Address
Hogskoleringen 1
Trondheim
SOBOLEV INSTITUTE OF MATHEMATICS SIBERIAN BRANCH OF RUSSIAN ACADEMY OF SCIENCES
Russia
Address
Pr T Academika Koptyuga 4
Novosibirsk
UNIVERSITA DEGLI STUDI DI TRENTO
Italy
Address
Via Belenzani 12
Trento
UNIVERSITY OF JYVÄSKYLÄ
Finland
Address
Seminaarinkatu 15
Jyväskylä
UNIVERSITÄT BERN
Switzerland
Address
Hochschulstrasse 4
Bern