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Periodic Report Summary 1 - MANET (Metric Analysis for Emergent Technologies)

The project objectives
MAnET is a Marie Curie Initial Training Network (ITN) devoted to the training of young researchers on new frontier of mathematics and its applications. The scientific objective of the project is to develop new and highly sophisticated instruments of metric analysis with applications to a large spectrum of emergent technological fields from human vision, and medical imaging to traffic dynamics, and robot design.
Metric analysis, allows to reconsider differential problems, in rich geometrical setting, non isotropic or non regular. Non isotropic geometrical settings, called sub-Riemannian arise while describing the motion of a system in which some directions are not allowed by a constraint, as models of the visual cortex, robotics, or traffic dynamic. Non regular metric analogue of these concepts arise as limits of regular surfaces, or minima of a functional. The differential instruments are no more sufficient to handle these objects and have to be replaced by instruments of geometric measure theory: mass transportation, and currents. These results can be naturally translated into efficient models. One of the most fascinating topic at the frontier between geometric measure theory and PDE is the theory of Soap films and minimal surfaces. Geometric analysis in Lie groups provides an elegant tool for modeling the visual cortex with its modular structure and provide Brain-inspired models of computer vision and robotics. The abstract instruments of optimal transport find natural applications to design neural network and traffic simulation or eye tracking.

The consortium
The consortium consists of 9 European University and 4 associated partners, an alliance of careful selected partners with a high reputation in a set of complementary disciplines consisting in Geometric Measure theory Subriemannian PDE, Mathematical modelling in geometrical setting, Neuroscience and Robotics.
The consortium is coordinated by the Alma Mater studiorum University of Bologna. Full Partners are Universities of Bern, University Autonoma of Barcelona, University of Helsinki, University of Jvaskula, University of Paris Sud, University of Trento, Technishe U. of Eindhowen and CNRS (Paris). These are located at leading European Universities with high level of research experience and established doctoral programs.
The associated partners are TSS a company leader in traffic simulation systems, I-Optics, a leading pioneer in diagnosis solutions and retinal scanner imaging systems and INAIL, one of most important prosthetic centers in Europe, and the ophthalmology department of the University R. Decartes.
We recruited 15 researchers: 4 experienced researchers, and 11 PdD students, who are trained in this interdisciplinary endeavor and develop the scientific work.

The Training activity
Local training courses The 9 participating research teams are High level Universities who offer high level structured courses on annual basis. Additional courses were offered ad hoc for the thematic training of the fellows. Two private and two clinical partners provided more applied training.
The network wide activity Two general conferences and 5 thematic workshop, were organized in the first two year of the project with participation of 2 visit researchers. The fellows of the consortium had the possibility to be exposed to a challenging frontier problems, to meet the best experts in their field and to present their work.
Training through research activity ESRs appointed in the project has been trained through research by an highly qualified senior academics who meet the student on a periodic basis.

The scientific results
The main results obtained with geometrical measure theory instruments are well-posedness of transport equations in the setting of vector fields with unbounded divergence. Subriemannian PDE were exploited to obtain regularity and stability for minimal surfaces in the Heisenberg group. A Steiner formula has been established, which can be considered a first result for facing more general curvature equations.
The problems, well known in the Euclidean setting, were still open
The visual cortex has been recently modelled as a subriemannian structure with a subriemannian metric, so that these instruments have been applied to models of vision, retinal vessel detection. We proposed a cortical based model of perceptual completion, based on minimal surfaces and subriemannian geodesics and started analysing retinal vessel images.

We collect here only the references of a few papers already accepted for publication on peer reviewed journals and refer for the complete list of publications and preprint to the project web site.
A. Clop, R. Jiang, J. Mateu, J. Orobitg, Linear transport equations for vector fields with sub-exponentially integrable divergence, Calc. Var. and PDE 55, 1 (2016).
R. Jiang, Y.Y. Liang, D.C. Yang, Weak Musielak-Orlicz Hardy Spaces and Applications Math. Nachr, published online: DEC 2015
M. Galli, M. Ritoré, Regularity of $C^1$ surfaces with prescribed mean curvature Calc. Var. and PDE 54,3, (2015), 2503-2516.
M. Galli, M. Ritoré, Area-stationary and stable surfaces of class C^1 in the sub-Riemannian Heisenberg group H^1, Math, 285 (2015), 737–765.
Z. M. Balogh, F. Ferrari, B. Franchi, E. Vecchi, K. Wildrick, Steiner's formula in the Heisenberg group, Nonlin. Anal. 126 (2015), 201-17.
E. J. Bekkers, R. Duits, A. Mashtakov, G.R.Sanguinetti (*joint main authors), A PDE Approach to Data-driven Sub-Riemannian Geodesics in SE(2), SIAM Journal on Im. Sci. 8, n. 4 (2015), pp. 2740–2770.
A. Mashtakov, Yu. Sachkov, Superintegrability of Sub-Riemannian Problems on Unimodular 3D Lie Groups, Diff. Eq., 2015.

The expected final results and their potential impact
The expected final result of the project will be to develop deep mathematical instruments and theories which can be applied to the challenging problems posed the new emerging technological problems. Up to now different instruments have been settled down for each of these problems, coming from pure or applied science. Our unitary approach will allow to face mathematical problems, which cannot be solved a single instrument of differential, metric or of measure theory and at the same time will allow to study challenging technological problems.
Our fellows, trained in a higher inter-disciplinary endeavor can have better carrier opportunities, both in academia and in the private sector.

The project website
We refer to the project website for a more detailed description of the activities carried out in the project,
For every other information will free to contact

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