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Metric Analysis for Emergent Technologies

Final Report Summary - 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 astract instruments of optimal tranport find natural applications to design neural network and traffic simulation or eye tracking.
The structure of the work can be summarized as follows:
WP1 (geometric measure theory) and WP2 (subriemannian PDE) develop the theoretic part
WP3 (soap films and minimal surfaces) and WP4 (models of vision) develop mathematical models
WP5, WP6, WP7 and WP8 develop applications to traffic simulatino, retinal vessel detection, eye tracking and robotics
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 14 researchers: 3 experienced researchers, and 11 PdD students, who are trained in this interdisciplinary endehavior and deveolp 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 properties of transport equations in invariant spaces [1], of heat flow in metric spaces, and Geometric inequalities on Heisenberg groups [2]. As for PDE in Subriemannian setting, we studied properties of the quasilinear equations [3] and in curvature equations In particular a Gauss-Bonnet Theorem [4] and motion by curvature in the Heisenberg groups [5]. In this setting we also studied properties of minimal surfaces [6] and the Bernstein's Problem [7]. Subriemannian instruments were used in models of the visual cortex [8], retinal vessel detection [9], image analysis [10], and visual illusions [11], We collect here only the references of a few papers already accepted for publication on peerreviewed journals and refer for the complete list of publications and preprint to the project web site.
[1] 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).
[2] Z. Balogh, A. Kristály, K. Sipos Geometric inequalities on Heisenberg groups on Calculus of Variations and Partial Differential Equations Vol 57, Issue 2 2016 pp. 1-41 (article n° 61)
[3] S. Mukherjee, X. Zhong C^1,α -Regularity for variational problems in the Heisenberg group 2017 preprint
[4] Z. Balogh, J. T. Tyson, E. Vecchi Intrinsic curvature of curves and surfaces and a Gauss-Bonnet Theorem in the Heisenberg group on Mathematische Zeitschrift Vol 287 2016 pp. 1-38
[5] G. Citti, E. Baspinar Uniqueness of viscosity mean curvature flow solution in two sub-Riemannian structures 2016 preprint
[6] M. Galli, M. Ritoré, Regularity of $C^1$ surfaces with prescribed mean curvature Calc. Var. and
PDE 54,3, (2015), 2503-2516.
[7] E. Le Donne, S. Nicolussi Golo Regularity properties of spheres in homogeneous groups on Transactions of the American Mathematical Society Vol 370 2017 pp. 2057-20
[8] M. Favali, G. Citti, A. Sarti Local and global gestalt laws: A neurally based spectral approach on Neural Computation Vol 29 2017 pp. 394-422
[9] S. Abbasi-Sureshjani, M. Favali, G. Citti, A. Sarti, B. ter Haar Romeny Curvature Integration in a 5D Kernel for Extracting Vessel Connections in Retinal Images on IEEE Transactions on Image Processing Vol 27 2018 pp. 606 - 621
[10] A. Mashtakov, Yu. Sachkov, Superintegrability of Sub-Riemannian Problems on Unimodular 3D Lie
Groups, Diff. Eq. 2015.
[11] B. Franceschiello, A. Sarti, G. Citti A neuro-mathematical model for geometrical optical illusions on Journal of Mathematical Imaging and Vision Vol 60 2018 pp. 94-108

The final results and their potential impact
The final result of the project has been to develop deep mathematical instruments and theories which can be applied to the challenging problems posed the new emerging technological problems. Our fellows, trained in a hygher interdisciplinar endehavior can have optimal carrier opportunities: buy now 5 fellows completed their PhD: 1 found a permanent job in a private sector, and the other temporary works in academia.
The project website
We refer to the project website for a more detailed description of the activities carried out in the project,
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