Geometric and Harmonic Analysis with Interdisciplinary Applications

Project Information

GHAIA

Grant agreement ID: 777822

DOI

10.3030/777822

Project closed

EC signature date 2 October 2017

Start date 1 November 2017

End date 30 September 2023

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 1 930 500,00

EU contribution

€ 1 782 000,00

Coordinated by

ALMA MATER STUDIORUM - UNIVERSITA DI BOLOGNA
Italy

Periodic Reporting for period 2 - GHAIA (Geometric and Harmonic Analysis with Interdisciplinary Applications)

Reporting period: 2019-11-01 to 2023-09-30

Summary of the context and overall objectives of the project

GHAIA promoted excellence in pure and applied mathematical research through mobility to the worldwide leading Universities in the field. In USA, Princeton University, the University of California, the Massachusetts Institute of Technology (MIT), the University of Texas, Yale University, Johns Hopkins University, the University of Houston, the University of Pittsburgh, the Washington University (St. Louis), and the Worchester Polytechnical Institute have joined the consortium, together with the Academia Sinica at the University of Taipei, in Asia, and the University of Buenos Aires in South America. Two highly technological enterprises have joined the challenge: Marposs, a world leader in precision equipments for automatic inspection, and GMV, one of the main players in the implementation of the European strategy in the area of satellite navigation.

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The overall ambitious goal of obtaining a new unified framework of tools and results will be pursued by addressing a number of concrete tasks. The work is developed at the intersection of disciplines and will target a selection of challenging goals, each driving a dedicated WP:

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The latest scientific and technological developments are accompanied with an increasing need to apply advanced quantitative methods of mathematics, both in the natural sciences and in engineering. Deep theoretical results of mathematics, if correctly handled, have indeed proved to be able to give unexpected and effective answers to difficult applied problems.

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.

The perception of the Kanizsa image, reconstructed via propagation in the geometry of the cortex