#### Sharp Isoperimetric Inequalities – Old and New

Isoperimetric inequalities constitute some of the most beautiful and ancient results in geometry, going back to the story of the founding of Carthage by Queen Dido.

These inequalities play a key role in numerous facets of differential geometry, analysis, partial differential equations, calculus of variations, geometric measure theory, minimal surfaces, probability theory and more.

The isoperimetric variational problem is simple to state but often very hard to resolve: among all sets of prescribed volume in a given space, characterize those sets whose surface-area is minimal. For example, in Euclidean space, the Euclidean ball minimizes the surface-area among all sets of a given volume. Similarly, the isoperimetric minimizers on spherical, hyperbolic and Gaussian spaces have been determined and are nowadays classical.

The isoperimetric problem is well-understood on two-dimensional 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 Euclidean space into two bounded regions of enclosing prescribed volumes so that the total common surface-area is minimized, the solution is a double-bubble, as the ones we see in nature when two soap-bubbles attach. In Gauss space, a complete characterization of how to optimally partition the space into multiple regions was was recently established in our work with Joe 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 a 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.