From a player’s dream to mathematics: Carlos Andrés defends his thesis at IMPA

The first scientific laboratory of Carlos Andrés Toro Cardona was the sky. Before the equations and theorems came the stars – observed alongside his astronomy-loving uncle and revisited in the stories told by Carl Sagan in Cosmos, he saw with his mother, Glória. It was there that he saw his first mathematical demonstration: the proof of the irrationality of the square root of two. Little by little, the dream of being a footballer gave way to another: that of dedicating himself to scientific studies.

At the age of 30, Carlos Andrés is defending his thesis “Minimum surfaces of free edge in B³”, under the supervision of Professor Lucas Ambrozio. The work contributes to open questions in differential geometry, one of the most active areas of contemporary mathematics. The defense will take place on Tuesday (3), at 2pm in room 232, and will be broadcast by IMPA’s YouTube.

Born in Medellín, Colombia, Carlos Andrés spent his childhood in the municipality of Urrao, in the interior of Antioquia. At the time, he was still thinking about pursuing his dream of becoming a professional footballer. In high school, his career began to take shape with the influence of his chemistry teacher, Oscar Tulio.

“He taught me that the first step towards true knowledge is to accept the fact that we really don’t know. When we recognize our ignorance with courage, a genuine desire to investigate is born,” he says.

Driven by this spirit, he decided to devote a year to self-taught study before going to university. He read chemistry and physics books and immersed himself in differential and integral calculus. “I learned that mathematics was the best language and tool we have for understanding the laws of nature.” He went on to study physics at the University of Antioquia, where he came into contact with high-energy physics.

It was a competition for a scientific initiation scholarship in the Mathematics department that redefined his path. Under the guidance of Pedro Rizzo, a doctoral student at IMPA, he worked with the calculation of fundamental groups of classical groups – and had his first contact with a book by Elon Lages Lima, director-general of IMPA between 1969-1971; 1979-1989; e 1989-1993.

The desire to go to IMPA was born at this point. Accepted for a master’s degree under Professor Henrique Bursztyn, he faced the challenge of migrating from physics to mathematics. “It was very difficult to make the change from physical to mathematical thinking, but little by little I developed the ability to transform intuitive ideas into more rigorous arguments.”

During his doctorate, a course in Riemannian Geometry II with Professor Lucas Ambrozio set his course definitively: it would be in differential geometry that he would build his career.

In his thesis, Carlos Andrés investigates minimal surfaces – which are classic objects of differential geometry. Defined as critical points of the area functional, they are characterized by having zero mean curvature. In recent decades, a topic that has gained momentum in this field has been the study of so-called free-edge minimal surfaces. In addition to locally minimizing areas, these surfaces orthogonally intersect a fixed subvariety– for example, the unit sphere in the three-dimensional Euclidean ball.

In the orientable case, all the possible topologies have already been realized on the three-dimensional ball. But the scenario changes when it comes to non-orientable surfaces – those which, like the Möbius strip, only have one side. Thus, the main contribution of the thesis was to demonstrate that the Möbius strip cannot be realized as a minimal free-edge surface on the three-dimensional Euclidean ball.

“The Möbius strip has the simplest of non-orientable topologies. Showing that it can’t be realized in this context reveals an interesting geometric obstruction,” he explains.

The work goes further. Carlos Andrés investigated complete non-orientable minimal surfaces, of finite total curvature, whose ends are flanked by closed curvature lines – a necessary condition if such surfaces contain a portion of free edge on some ball. He showed that this is a rigid situation and that there are no examples with a single end.

The results advance the understanding of geometric limitations and possibilities in the non-orientable context, an area that remains open to question.

If in the master’s the challenge was to change language, in the doctorate it was something else: to produce new knowledge. “We went from watching the beauty of a great building to participating in its construction,” he says. “Professor Lucas Ambrozio’s guidance was fundamental. Every conversation was a source of inspiration.”

The first mathematician in his family, Carlos Andrés highlights the support he has received along the way. He is now preparing applications for post-doctoral studies in Brazil and abroad.