Abstract: In geometry and topology, as well as in applications of Mathematics to Physics and other areas, one often deals with a system of 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 condition is also sufficient. This led to counter-intuitive results, like Steven Smale’s famous inside-out "eversion" of the sphere 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 continue to be discovered.
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Abstract: I will discuss the origin and the history of symplectic topology, its basic problems, methods, and applications.
Abstract: Flexible and rigid methods coexisted in symplectic topology from its inception. Though overshadowed by the spectacular development of the rigid side, the flexible side of symplectic topology also had, and continues to have a remarkable success story, especially in the last 3 years. I will survey recent developments in my lecture.