01/10/2019

At IMPA, Roberto Velho researches mean-field games.

Cecilia Manzoni

What is the shortest route to get from home to work? How to choose the right products at the supermarket to reduce the cost of monthly shopping without sacrificing any necessities? How to minimize production costs in a factory? Or, how can we determine the minimum speed required for a rocket to escape Earth's gravitational pull?

These are some of the questions that optimization, an area of Applied Mathematics that develops methods to find the best available option for real-world problems, seeks to answer, according to established criteria. Motivated by the concreteness of this type of problem, Roberto Machado Velho dedicates himself to unraveling them in his postdoctoral research at IMPA, an institute he has been a part of since April.

An expert in the fields of analysis and partial differential equations, Roberto is one of the few Brazilians researching mean-field game theory. While there is no official translation into Portuguese, the term refers to the study of strategic decision-making in very large populations of small, interactive agents.

"Mean field games are problems of approximating a solution to a game when many players are trying to perform the same task optimally," the mathematician simplifies.

They have applications in the financial market and can influence aspects of the macroeconomy, such as the interest rates set by a country's government.

During his childhood in the bucolic Vila Isabel neighborhood in Rio's North Zone, this type of question already intrigued Roberto, sparking his interest in mathematics. It was the beginning of the 90s, and there wasn't a Brazilian who wasn't worried about the rampant inflation that reached over 80% per month.

“My siblings and I would go to the supermarket and compete to see who could do the math the fastest. Since product prices varied a lot back then, it was a real challenge,” recalls the postdoctoral researcher.

His mathematical talent began to blossom in elementary school, with his first contact with algebraic structures. Later, in high school, he studied at CEFET/RJ, where he took a technical course in Electrotechnics. He ended up entering the Electronic and Computer Engineering program at the Federal University of Rio de Janeiro.

He spent a few semesters in the department, but Roberto realized that the approach to mathematics in the course did not meet his needs. "I realized that if I graduated in electronic engineering, I wouldn't leave the institution mastering mathematics. I wanted to go further, to learn more."

Participation in a special calculus class with top-ranked students from the Engineering and Applied Mathematics courses put an end to his doubts. In 2004, the then-university student enrolled in Applied Mathematics at the same UFRJ (Federal University of Rio de Janeiro), and soon began to participate in scientific initiation projects in Computer Science.

When choosing a master's program, Roberto faced another dilemma between a more academic education and one more geared towards the market. After much deliberation, he discarded the master's program in Electrical Engineering at PUC-Rio to enroll in the master's program in Applied Mathematics at UFRJ. Supervised by Professor Fábio Antonio Ramos, Roberto defended his dissertation on ordinary differential equations and transport equations with Sobolev regularity in 2011.

At the end of that same year, the mathematician received an offer to join the doctoral program in Mathematics at the Instituto Superior Técnico in Lisbon. "At that time, I had already become interested in optimal control mechanisms, and Professor Diogo Gomes, one of the few researching the topic, invited me to work with him."

His time in Portugal was brief. After a year and a half, Gomes received an invitation to join the scientific staff of King Abdullah University of Science and Technology, a higher education institution that had recently opened in Saudi Arabia.

Despite the unusual destination, Roberto followed his mentor, embarking on a journey to Thuwal. The village, which used to be a quiet fishing colony, was chosen to house the immense 500,000 m² King Abdullah University campus.

The system of absolute monarchy, the arid climate, and the strict laws imposed by Sharia are some of the many factors that separate Brazilian culture from Saudi Arabian culture. For Roberto, the impact of these differences was mitigated. "The university had many foreigners, and within the campus, many rules are different from those in the country," he says.

Still, the experience of living in such a different country helped him to demystify stereotypes about that society. "They're not so different from the Western world," adds the mathematician, who traveled with friends to various villages in the country.

Living in an apartment with a balcony overlooking the Red Sea, Roberto had nothing to complain about. The university's facilities were impeccable, and the atmosphere fostered interaction with master's, doctoral, and post-doctoral students from various fields.

In his doctoral project, the mathematician discussed mean-field games with an emphasis on crowd motion problems, a subfield with practical applications such as building evacuation plans. At IMPA, Roberto continues to develop research on the topic and aims to disseminate it throughout the country.

Located in the Tijuca Forest, the institute is an environment that inspires him in this sense. "Here we have a unique setting, a pleasant and quiet space, a good reading room, and a scientific body composed of researchers from different areas of Mathematics, with whom I can dialogue and learn about other research," he says.

With a solid career in Applied Mathematics, the postdoctoral researcher hasn't ruled out working in the market in the future. "It depends on the opportunities that arise. If something comes up in the industry with interesting problems that appeal to me, it's an option."