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CHallenges in ANalysis and GEometry between mean and scalar curvature

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 striki... ver más

28/02/2027

UNITN

1M€

Presupuesto del proyecto: 1M€

Líder del proyecto

UNIVERSITA DEGLI STUDI DI TRENTO No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Fecha límite participación Sin fecha límite de participación.

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Financiación concedida El organismo H2020 notifico la concesión del proyecto el día 2020-09-14

ERC-2020-STG: ERC STARTING GRANTS Scope:Objectives

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3 Participantes

UNITN

955.00K€ | Lider

FON LABS

149.72K€ | Participante

ETH Zürich

387.50K€ | Participante

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Duración del proyecto: 77 meses Fecha Inicio: 2020-09-14
Fecha Fin: 2027-02-28

Líder del proyecto

UNIVERSITA DEGLI STUDI DI TRENTO No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 1M€

Fecha límite de participación Sin fecha límite de participación.

Descripción del proyecto 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.

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