Summary
The unifying goal of the CURVATURE project is to develop new strategies and tools in order to attack fundamental questions in the theory of smooth and non-smooth spaces satisfying (mainly Ricci or sectional) curvature restrictions/bounds.
The program involves analysis and geometry, with strong connections to probability and mathematical physics. The problems will be attacked by an innovative merging of geometric analysis and optimal transport techniques that already enabled the PI and collaborators to solve important open questions in the field.
The project is composed of three inter-connected themes:
Theme I investigates the structure of non smooth spaces with Ricci curvature bounded below and their link with
Alexandrov geometry. The goal of this theme is two-fold: on the one hand get a refined structural picture of
non-smooth spaces with Ricci curvature lower bounds, on the other hand apply the new methods to make progress in some long-standing open problems in Alexandrov geometry.
Theme II aims to achieve a unified treatment of geometric and functional inequalities for both smooth and
non-smooth, finite and infinite dimensional spaces satisfying Ricci curvature lower bounds. The approach
will be used also to establish new quantitative versions of classical geometric/functional inequalities for smooth Riemannian manifolds and to make progress in long standing open problems for both Riemannian and sub-Riemannian manifolds.
Theme III will investigate optimal transport in a Lorentzian setting, where the Ricci curvature plays a key
role in Einstein's equations of general relativity.
The three themes together will yield a unique unifying insight of smooth and non-smooth structures with curvature bounds.
The program involves analysis and geometry, with strong connections to probability and mathematical physics. The problems will be attacked by an innovative merging of geometric analysis and optimal transport techniques that already enabled the PI and collaborators to solve important open questions in the field.
The project is composed of three inter-connected themes:
Theme I investigates the structure of non smooth spaces with Ricci curvature bounded below and their link with
Alexandrov geometry. The goal of this theme is two-fold: on the one hand get a refined structural picture of
non-smooth spaces with Ricci curvature lower bounds, on the other hand apply the new methods to make progress in some long-standing open problems in Alexandrov geometry.
Theme II aims to achieve a unified treatment of geometric and functional inequalities for both smooth and
non-smooth, finite and infinite dimensional spaces satisfying Ricci curvature lower bounds. The approach
will be used also to establish new quantitative versions of classical geometric/functional inequalities for smooth Riemannian manifolds and to make progress in long standing open problems for both Riemannian and sub-Riemannian manifolds.
Theme III will investigate optimal transport in a Lorentzian setting, where the Ricci curvature plays a key
role in Einstein's equations of general relativity.
The three themes together will yield a unique unifying insight of smooth and non-smooth structures with curvature bounds.
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More information & hyperlinks
| Web resources: | https://cordis.europa.eu/project/id/802689 |
| Start date: | 01-02-2019 |
| End date: | 30-04-2025 |
| Total budget - Public funding: | 1 256 221,00 Euro - 1 256 221,00 Euro |
Cordis data
Original description
The unifying goal of the CURVATURE project is to develop new strategies and tools in order to attack fundamental questions in the theory of smooth and non-smooth spaces satisfying (mainly Ricci or sectional) curvature restrictions/bounds.The program involves analysis and geometry, with strong connections to probability and mathematical physics. The problems will be attacked by an innovative merging of geometric analysis and optimal transport techniques that already enabled the PI and collaborators to solve important open questions in the field.
The project is composed of three inter-connected themes:
Theme I investigates the structure of non smooth spaces with Ricci curvature bounded below and their link with
Alexandrov geometry. The goal of this theme is two-fold: on the one hand get a refined structural picture of
non-smooth spaces with Ricci curvature lower bounds, on the other hand apply the new methods to make progress in some long-standing open problems in Alexandrov geometry.
Theme II aims to achieve a unified treatment of geometric and functional inequalities for both smooth and
non-smooth, finite and infinite dimensional spaces satisfying Ricci curvature lower bounds. The approach
will be used also to establish new quantitative versions of classical geometric/functional inequalities for smooth Riemannian manifolds and to make progress in long standing open problems for both Riemannian and sub-Riemannian manifolds.
Theme III will investigate optimal transport in a Lorentzian setting, where the Ricci curvature plays a key
role in Einstein's equations of general relativity.
The three themes together will yield a unique unifying insight of smooth and non-smooth structures with curvature bounds.
Status
SIGNEDCall topic
ERC-2018-STGUpdate Date
27-04-2024
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