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CHallenges in ANalysis and GEometry, between mean and scalar curvature

Descrizione del progetto

Indagini matematiche all’avanguardia

La geometria euclidea è lo studio dello spazio piatto, la geometria con cui tutti abbiamo maggiore familiarità. Le sue idee si possono spiegare disegnando su un foglio di carta. Al contrario, la geometria riemanniana è, più in generale, l’area della matematica che riguarda lo studio delle forme attraverso il concetto di curvatura. Il suo linguaggio e i suoi metodi si sono dimostrati estremamente versatili e sono impiegati per descrivere un’ampia gamma di fenomeni naturali, comprese le forze gravitazionali nel contesto della relatività generale di Einstein. Il progetto CHANGE, finanziato dall’UE, è costruito intorno a una rete di problemi negli ambiti della matematica di base e applicata, e si propone di chiarire ulteriormente alcune questioni centrali nel campo.

Obiettivo

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.

Meccanismo di finanziamento

ERC-STG - Starting Grant

Istituzione ospitante

UNIVERSITA DEGLI STUDI DI TRENTO
Contribution nette de l'UE
€ 955 000,00
Indirizzo
VIA CALEPINA 14
38122 Trento
Italia

Mostra sulla mappa

Regione
Nord-Est Provincia Autonoma di Trento Trento
Tipo di attività
Higher or Secondary Education Establishments
Collegamenti
Costo totale
€ 955 000,00

Beneficiari (2)