Symplectic geometry: applications and challenges (Q6581221)

The article intents to give some brushstrokes of symplectic geometry for a general (non-mathematical) public without trivializing the ideas. The author starts with a very rapid history of geometry and explains in general terms the difference between differential and algebraic geometry and that symplectic geometry originated in the study of planetary movements. A section on ``milestones of symplectic geometry'' surveys the Poincaré-Birkhoff theorem, the Arnold conjecture, Gromov's nonsqueezing theorem, and the convexity of moment maps due to \textit{M. F. Atiyah} [Bull. Lond. Math. Soc. 14, 1--15 (1982; Zbl 0482.58013)] and \textit{V. Guillemin} and \textit{S. Sternberg} [Invent. Math. 67, 491--513 (1982; Zbl 0503.58017)]. Another section describes quite a few applications of symplectic geometry to integrable dynamical systems. Among the latter are the periodic ones (``toric systems''), for which the author describes in some detail the work of \textit{T. Delzant} [Bull. Soc. Math. Fr. 116, No. 3, 315--339 (1988; Zbl 0676.58029)] proving that an associated polytope is a complete invariant of toric integrable dynamical systems and mentions his own work with \textit{S. Vũ Ngọc} [Invent. Math. 177, No. 3, 571--597 (2009; Zbl 1215.53071)], thus generalizing this to semitoric systems. The mathematical formalisms of differential forms and symplectic manifolds and their local and global properties are informally explained in an appendix.