Multi-point maximum principles, integrability profiles, and smoothing

Regularity theory forms the foundation of differential equations governing the behaviour of systems such as gravitation, climate modelling, biological systems, economics models, geometric structures such as minimal surfaces and many others. Tremendous developments occurred during the 20th century, yet many unresolved questions remain (such as the Millennial prize problem on the regularity of Navier-Stokes). We propose new methods to tackle such questions via a geometric approach to obtain optimal inequalities. The project will greatly simplify the very technical aspects of the field and open up new avenues to solve the major outstanding problems in this area.