Min-max constructions in geometry and topology – Min-Max

This collaborative research project aims to bring together researchers from various areas - namely, geometry and topology, minimal surface theory and geometric analysis, and computational geometry and algorithms - to work on a precise theme around min-max constructions and waist estimates.

These past few years, min-max techniques led to groundbreaking advances in the field of geometry, including the resolution of the Willmore conjecture and the Yau conjecture, using an approach developed by Marques and Neves. These breakthroughs combine analytical techniques from the Almgren-Pitts
theory ensuring the existence of minimal hypersurfaces through min-max arguments and recent min-max estimates of Gromov and Guth based on topological considerations. Relying on different branches of geometry, analysis and topology, the resolution of these conjectures opened a new chapter of differential
geometry by promoting the min-max theory and its applications, as Perelman's resolution of Poincaré's conjecture opened a new chapter with the Ricci
flow.

The central theme of this proposal is to study the geometry and topology of geometrical objects through Morse theory min-max processes on the space of cycles for various functionals measuring the size of the cycles. A special focus will be given to the implementation of new geometric constructions effective enough to lead to the development of algorithms in computational geometry.

In the description of the project, we set forth three largely overlapping themes about minimal surface theory, quantitative homotopy theory, and combinatorial and non-combinatorial topology:

(1) Minimal surface theory,
(a) Index estimates, topology and classification,
(b) Discrete surfaces in 3-manifolds and applications;

(2) Quantitative homotopy theory,
(a) Sweepout estimates in Riemannian geometry,
(b) Pants decomposition,
(c) Arborescent sweepouts, embedded graphs and algorithms;

(3) Combinatorial and non-combinatorial topology,
(a) Leray's acyclic cover theorem and the Kalai-Meshulam projection theorem,
(b) From the selection lemma to waist theorems and back.

This project calls for close collaboration between researchers from three scientific communities since progress in one area can often be adapted to solve problems in another area as illustrated by some of the questions raised in this proposal. By joining the complementary expertise of our consortium, our goal is to develop this min-max approach from different perspectives, making our project extremely coherent around this fast-growing topic.