Talk: Flexible Mathematics: Introduction to the h-principle
Abstract: In geometry and topology, as well as in applications of Mathematics to Physics and other areas, one often deals with system of (partial) differential equations and inequalities. Given such a system, by replacing derivatives of unknown functions by independent functions one gets a system of algebraic equations and inequalities. Of course, the solvability of the algebraic system is a necessary condition for solvability of the original system of differential equations. It was a surprising discovery in 1950-60s that there are geometrically interesting classes of examples of systems for which this condition is also sufficient. Usually this led to counter-intuitive results, like famous Steven Smale’s inside-out eversion of the 2-sphere in the 3-space, or John Nash’s isometric (i.e. preserving lengths of all curves) embedding of the unit sphere into a ball of an arbitrary small radius. Since that time many more examples of this phenomenon were discovered and continue to be discovered.
In the talk I will discuss the history of this subject, some underlying ideas, as well as its modern development and applications.
In 2016 Yakov Eliashberg was awarded the Crafoord Prize in Mathematics from the Swedish Academy of Sciences for the development of contact and symplectic topology and groundbreaking discoveries of rigidity and flexibility phenomena.