Barcelona Graduate School of Mathematics (BGSMath)

In September 2017, UPC Associate Professor and BGSMath Faculty member Marcel Guàrdia, 35, received an ERC Starting Grant. The European Research Council carefully selects talented early-career scientists who are ready to work independently and awards them up to 1.5 million euros for a 5-year period. His ERC project focuses on “Instabilities and homoclinic phenomena in Hamiltonian systems”. Guardia’s Starting Grant is one of 11 such grants obtained in Spain, and the only one in mathematics.

Marcel’s scientific interests are mainly dynamical systems. “I study phenomena that evolve with time,” he explains. “It’s a field that pulls lots of strings: this is why I chose it in the first place.” Marcel’s areas of expertise are differential equations and Hamiltonian systems. Hamiltonian mechanics is a reformulation of classical mechanics that provides a more general understanding of the theory.

During the past years, Marcel applied his studies to the classical 3-body problem in celestial mechanics, a problem over which mathematicians have puzzled for more than three centuries. The question is how three objects orbit one another according to the classic Newton’s laws of dynamics.

Together with his colleagues, he showed that in the majority of the cases, the three bodies can have oscillatory motions. Technically, he explains, the superior limit position is infinity, and the lower limit is a real number. In practice, the position of one of the planets oscillates between infinity and a region of the space, which – in physical terms – means that a planet can go as far as one wants, but it will always come back. This is called a “comet-like behavior.”

The most important article that convinced the ERC’s panel appeared in *Inventiones Mathematicae* in 2016. In it, Marcel Guàrdia and his co-authors studied a very special type of behavior of a restricted 3-body problem: a small mass taken to simplify equal to zero under the influence of the Newtonian gravitational force exerted by two massive bodies (for instance, the Sun and Jupiter).

Over the next years, and thanks to the ERC funding, Marcel will work on the dynamics of the N-body problem, looking for instabilities, while also extending his research on Schrödinger’s equation instability to other more general situations.

“My most important working tools? They are called invariant manifolds, they are like highways, along which unstable behaviours travel,” he explains with a metaphor. Then he gives a practical example of how this field of mathematics is used in real life: “Space agencies use these techniques to calculate spaceships’ trajectories to minimise fuel usage,” he says.