The research that the CAREER Award will support belongs to the context of dynamics and geometry. Dynamics is the study of the motion of bodies. Geometry is the study of shape (broadly understood) of objects and spaces.

More specifically, the CAREER Award will support Pelayo's research at the intersection of symplectic geometry, spectral theory and dynamics. Symplectic geometry has its roots in physics, and provides an appropriate mathematical framework to study many problems of physics and chemistry and their quantum counterparts.

Pelayo's research on this topic exhibits the interplay between mathematical and physical theories. Indeed, several groups of physicists and chemists working on modern quantum spectroscopy have been interested in seeing how mathematical methods can contribute to advance their research, and predict new physical phenomena. They have been particularly interested in understanding the global structure of joint energy-momentum spectra of small molecules.

Even more, they have recognized the pivotal role that mathematical invariants play in this problem. Their works have motivated a large number of mathematical questions. The physicists have asked whether one can single out an optimal set of mathematical invariants that would characterize a physical system and then detect these invariants in the spectrum of the system. The detection of the invariants in the system spectrum will allow us to reconstruct the system and hence predict new phenomena. The CAREER Award will support Pelayo's investigations into this crucial question.

These applications to quantum molecular spectroscopy are integrated in Pelayo's approach of employing methods from pure mathematics (symplectic and spectral theory, microlocal analysis) to address problems from the applied sciences. More generally, Pelayo's research is focused on a fundamental type of physical system, the so called integrable systems.

One can find direct applications of integrable systems in numerous contexts. Some such examples are nonlinear control, locomotion generation in robotics, elasticity theory, plasma physics, field theory and planetary mission design.