Abstract: In geometry and topology, as well as in applications of Mathematics to Physics and other areas, one often deals with systems of partial differential equations and inequalities. By replacing derivatives of unknown functions by independent functions one gets a system of algebraic equations and inequalities. The solvability of this algebraic system is necessary for the solvability of the original system of differential equations. It was a surprising discovery in the 1950-60s that there are geometrically interesting classes of systems for which this necessary condition is also sufficient. This led to counter-intuitive results, like Steven Smale’s famous inside-out “eversion” of the 2-sphere and John Nash’s isometric embedding of the unit sphere into a ball of an arbitrary small radius. Many more instances of this phenomenon, called by Mikhail Gromov h-principle (“h” stands for homotopy), were since then found and continue to be discovered, most recently in symplectic topology.
I will describe the main ideas and classical methods and results of the h-principle.

Coffee will be served at 12:00 before the lecture
at Schreiber building 006