Álvaro Pelayo

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

 Summary of G1 : Hamiltonian Dynamics and Symplectic Geometry

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:

• Symplectic Capacities & Embeddings:

studying the existence or inexistence of symplectic embeddings of balls and cylinders into balls and cylinders, following Gromov, Polterovich, Guth, Hind-Kerman. These results have consequences in terms of symplectic capacities. They may also be interpreted terms of stability (or instability) of Hamiltonian differential equations, following the works of Kuksin and Bourgain in the 1990s. I gave talks on this viewpoint which is perhaps less known in geometry at the PDE Seminar in Austin, and at the conference honoring Alan Weinstein in July 2013 (I'm preparing notes to post here);

• Random Dynamical Systems:

the interaction between random and deterministic ideas in dynamics is a highly active topic of research, see for instance the recent work of Bourgain-Sarnak-Ziegler (2013), and Sarnak's Lectures (2011) on "Mobius function Randomness and Dynamics". I have studied the generalization of the Poincare-Birkhoff Fixed Point Theorem to area-preserving twist maps that are random with respect to a given probability measure. While random dynamics has been explored quite throughly, eg. Brownian motions, the implications of the area-preservation assumption remain relatively unknown. Currently I continue working on random versions of theorems in symplectic geometry using a theory of random generating functions introduced jointly with Rezakhanlou, following Chaperon and Viterbo;

• Semitoric Systems:

using symplectic methods to construct invariants of finite-dimensional integrable systems of so called semitoric type: these are systems with 2 degrees of freedom on 4-dimensional manifolds and for which one of the Hamiltonians generates a periodic motion. Semitoric systems retain some of the rigidity of toric systems but may have in addition nodal singularities (corresponding to singular fibers which are multipinched tori). Semitoric systems model some simple physical systems such as the spherical pendulum or the coupled spin-oscillator (known as Jaynes-Cummings model in quantum optics);

• Singular Affine Structures:

studying singular affine structures induced by singular Lagrangian fibrations into the plane, and connecting them to the works of Gross and Siebert on mirror symmetry, the work of Symington on the symplectic topology of almost toric manifolds, and the work of the Fomenko school on topological classifications of integrable systems;

• Moduli Spaces:

endowing certain spaces from symplectic and algebraic geometry with geometric structures, such as symplectic or metric structures. With the aid of these structures, one can raise new questions about maps or other objects defined on these spaces (for example symplectic capacities), for which regularity questions may now be posed. In some cases these structures lead us to constructing new invariants.