Periodic Reporting for period 2 - GHAIA (Geometric and Harmonic Analysis with Interdisciplinary Applications)
Berichtszeitraum: 2019-11-01 bis 2023-09-30
We developed a new integrated framework of geometric and harmonic analysis, which unify in a original picture problems strongly influenced by symmetries and coming from different mathematical and technological fields. In this way we were able to contribute to some of the biggest challenges of this century: data analysis, models of brain functionality, human and artificial vision.
The final goal has been to obtain new geometrical instruments of machine vision, to be applied for measurement of industrial processes.
WP1 Representations and Lie theory develops a theory of infinite dimensional representations of Lie supergroups
WP2 Harmonic analysis and approximation theory develops a theory of sampling, dimensionality reduction and approximations in noncommutative and anisotropic settings
WP3 Degenerate, nonlinear and nonlocal PDE develops a theory of nonlinear propagation in degenerate and nonlocal setting
WP4 Geometric Analysis develops a new approach to the isoperimetric problem and to geometric PDE in Lie groups
WP5 Data, brain and vision introduces new models for the neural connectivity, eye tracking, image analysis, and general unsupervised learning of invariances
WP6 Industrial applications of Machine vision study automatic systems of artificial visual inspection for industrial production and of satellite images feature extraction for space navigation
The work was carried out through a large number of secondments, from the European partners to US partners and from Buenos Aires to to European partners. The main results gave rise to preprints, submitted for publication in the top impact ranked journals and to a large number of conferences in the different sectors, in very prestigious Universities.
We have started bridging the gap between the most elegant theoretical results in harmonic analysis, geometric PDE from one side, and some of the most exciting scientific and technological challenges of this century from the other: models of the human brain, natural and artificial vision, industrial applications to the design of new precision measuring instruments and new systems of satellite guidance.
Impact
The large number of secondments performed by researchers enrolled in a Ph.D. or in the first year after getting their Ph.D. degree gave them the opportunity to visit a prestigious university or to an important,research-focused, private enterprise. These secondments greatly help them in a crucial moment for their research career, when they have to make vital choices for their future. Senior reserachers from academia or private partners, had the important opportunities to strengthen the established collaborations and to start new ones with some of the highest ranked universities in the world.
Dissemination has been performed with publication in the top impact ranked journals and participation to a large number of conferences in the different sectors. In addition we reached a larger pubblic with press release and presentation of the results of the project beyond the project's own community, with many pubblich events. This will be a unique occasion to change the perception of mathematics in the large public, to create awareness about the research work performed and its implications for citizens.