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Abstract

Abstract. Flexible and rigid methods coexisted in symplectic topology from its inception. While the rigid methods dominated the development of the subject during the last three decades, the balance has somewhat shifted to the flexible side in the last three years. In the talk we survey the recent advances in symplectic flexibility in the work of S. Borman, K. Cieliebak, T. Ekholm, E. Murphy, I. Smith, and the author.This survey article is an expanded version of author's article [27] and his talk at the Current Events Bulletin at the AMS meeting in Baltimore in January 2014. It also uses materials from the papers [6,9,20,32] and the book [8]. The h-principleMany problems in mathematics and its applications deal with partial differential equations, partial differential inequalities or, more generally, with partial differential relations (see [48] and also [31]), i.e., any conditions imposed on partial derivatives of an unknown function. A solution of such a partial differential relation R is any function which satisfies this relation.With any differential relation one can associate the underlying algebraic relation by substituting all the derivatives entering the relation with new independent functions. The existence of a solution of the corresponding algebraic relation, called a formal solution of the original differential relation R, is a necessary condition for the solvability of R. Though it seems that this necessary condition should be very far from being sufficient, it was a surprising discovery in the 1950s of geometrically interesting problems where existence of a formal solution is the only obstruction for the genuine solvability. Two of the first such non-trivial examples were the C 1 -isometric embedding theorem of J. Nash and N. Kuiper [57,68] and the immersion theory of S. Smale and M. Hirsch [55,77]. After Gromov's remarkable series of papers, beginning with his paper [47] and culminating in his book [48], the area crystallized as an independent subject, called the h-principle.Rigid and flexible results coexist in many areas of geometry, but nowhere else do they come so close to each other, as in symplectic topology, which serves as a rich source of examples on both sides of the flexible-rigid spectrum. Flexible and rigid problems and the development of each side toward the other shaped and continue to shape the subject of symplectic topology from its inception.