Álvaro Pelayo                                                    

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A theory is more impressive the greater the simplicity of its premises, the more kinds of things it relates, and the more extended its area of applicability.

A. Einstein



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Summary of G1 : Hamiltonian Dynamics and Symplectic Geometry


pinchedtorus


In 1985 Gromov revolutionatized symplectic geometry by introducing pseudoholomorphic curves and using them to prove the celebrated Symplectic Nonsqueezing theorem: you cannot squeeze a big ball into an infinite thin cylinder by means of a symplectic map, or a Hamiltonian flow (even though the cylinder has infinite volume).

Gromov
proved this result by studying properties of moduli spaces of pseudoholomorphic curves. Gromov's Nonsqueezing result may also be interpreted as a strong instability result for Hamiltonian differential equations. Coming from the variational theory of Hamiltonian dynamics, Ekeland and Hofer gave a proof of Gromov's Nonsqueezing Theorem by studying periodic solutions of Hamiltonian systems. Hofer and Ekeland-Hofer introduced new invariants of symplectic manifolds, the so called symplectic capacities.

The unifying theme of G1 is the use of analytic and dynamical techniques to construct symplectic invariants of manifolds endowed with some interesting additional structure on it such as an integrable system, a Lagrangian foliation, a family of random maps, a circular symmetry, ..... The invariants I have constructed come primarily from studying the dynamics of Hamiltonian flows, singular affine structures, and normal forms of singularities; I have also used pseudoholomorphic curves and ergodic theory.

Topics I have worked on:

Research