Descripción del proyecto
Cerrar el círculo de una de las cuestiones más antiguas de la geometría
Los problemas isoperimétricos se remontan a la historia de la fundación de Cartago, cuando la reina Dido quiso delimitar una superficie de tierra lo suficientemente grande como para construir una ciudad entera con una sola piel de buey. En un espacio determinado, el problema isoperimétrico busca caracterizar las formas del volumen prescrito cuya superficie sea mínima. Por ejemplo, los antiguos griegos ya sabían que, de todos los conjuntos de un plano que delimitan una superficie determinada, el círculo tiene un perímetro mínimo. El problema se entiende perfectamente en las superficies bidimensionales, pero se vuelve mucho más complejo y difícil al aplicarse en tres o más dimensiones. El proyecto ISOPERIMETRY, financiado con fondos europeos, prevé abordar estos retos en varios entornos importantes y naturales mediante nuevas herramientas innovadoras.
Objetivo
Isoperimetric inequalities constitute some of the most beautiful and ancient results in geometry, and play a key role in numerous facets of differential geometry, analysis, calculus of variations, geometric measure theory, minimal surfaces, probability and more.
Isoperimetric minimizers have classically been determined on Euclidean, spherical, hyperbolic and Gaussian spaces. The isoperimetric problem is well-understood on surfaces, but besides some minor variations on these examples and some three-dimensional cases, remains open on numerous fundamental spaces, like projective spaces, the flat torus or hypercube, and for symmetric sets in Gaussian space. When partitioning the space into multiple regions of prescribed volume so that the common surface-area is minimized, the Euclidean double-bubble conjecture was established by Hutchings-Morgan-Ritoré-Ros, and the Gaussian multi-bubble conjecture was recently established in our work with Neeman, but the Euclidean and spherical multi-bubble conjectures remain wide open. Isoperimetric comparison theorems like the Gromov-Lévy and Bakry-Ledoux theorems are well-understood under a Ricci curvature lower bound, but under an upper-bound K ≤ 0 on the sectional curvature, the Cartan-Hadamard conjecture remains open in dimension five and higher despite recent progress. In the sub-Riemannian setting, the isoperimetric problem remains open on the simplest example of the Heisenberg group.
The above long-standing problems lie at the very forefront of the theory and present some of the biggest challenges on both conceptual and technical levels. Any progress made would be extremely important and would open the door for tackling even more general isoperimetric problems. To address these questions, we propose adding several concrete new tools, some of which have only recently become available, to the traditional ones typically used in the study of isoperimetric problems.
Ámbito científico
Programa(s)
Régimen de financiación
ERC-COG - Consolidator GrantInstitución de acogida
32000 Haifa
Israel