Project description DEENESFRITPL Mathematical investigations that are ahead of the curve Euclidean geometry is the study of flat space – the geometry we are all the most familiar with. Its ideas can be explained with drawings on a flat piece of paper. Riemannian Geometry is instead, more generally, the area of mathematics that concerns the study of shapes through the concept of curvature. Its language and methods have proven to be extremely versatile and are employed to describe a large range of natural phenomena, including gravitational forces in the context of Einstein's general relativity. The EU-funded CHANGE project is built around a network of problems in the realms of both basic and applied mathematics, aiming to shed new light on some central questions in the field. Show the project objective Hide the project objective Objective The interplay between Analysis and Geometry has led to a number of spectacular achievements such as the proof of the Poincaré conjecture by Perelman. The goal of this proposal is to establish a research group that will make striking progress along the following two directions, that reflect the two souls, extrinsic and intrinsic, of Riemannian Geometry, as well as their mutual interaction. Minimal surfaces, namely surfaces of zero mean curvature, have been an object of mathematical study since the 18th century (with pioneering work by Lagrange and Euler), and yet remain at the heart of many problems to this day. I aim at shading new light on their understanding, by means of a thorough investigation of the Morse index as an observable on the space of minimal cycles, both in general 3-manifolds of positive curvature and in space forms, towards higher Urbano-type theorems and beyond min-max techniques. Partly motivated by the study of data sets for the Einstein equations on the one hand, and by a far-reaching program by Gromov on the other, we also want to systematically study the interplay between the scalar curvature of a manifold and the mean curvature of its boundary. The project, which builds on my recent contributions and long-term experience in the field, relies on a combination of diverse elliptic and parabolic techniques, and aims at developing effective deformation methods that will have a variety of applications.These directions, while seemingly different, are deeply intertwined both at the technical and conceptual level, and incarnate the primary goal of redefining the state of the art in the investigation of infinite-dimensional spaces of solutions to fundamental geometric problems. Fields of science natural sciencesmathematicspure mathematicsgeometry Programme(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Topic(s) ERC-2020-STG - ERC STARTING GRANTS Call for proposal ERC-2020-STG See other projects for this call Funding Scheme ERC-STG - Starting Grant Host institution UNIVERSITA DEGLI STUDI DI TRENTO Net EU contribution € 955 000,00 Address VIA CALEPINA 14 38122 Trento Italy See on map Region Nord-Est Provincia Autonoma di Trento Trento Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 955 000,00 Beneficiaries (2) Sort alphabetically Sort by Net EU contribution Expand all Collapse all UNIVERSITA DEGLI STUDI DI TRENTO Italy Net EU contribution € 955 000,00 Address VIA CALEPINA 14 38122 Trento See on map Region Nord-Est Provincia Autonoma di Trento Trento Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 955 000,00 EIDGENOESSISCHE TECHNISCHE HOCHSCHULE ZUERICH Switzerland Net EU contribution € 387 500,00 Address Raemistrasse 101 8092 Zuerich See on map Region Schweiz/Suisse/Svizzera Zürich Zürich Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 387 500,00
