New generation of Brazilian mathematicians gathers at São Paulo conference

Fields Medal winner Artur Ávila attends the conference at the University of São Paulo’s Mathematics & Statistics Institute.

By Diego Freire

Agência FAPESP – In 2014, Brazil won its first Fields Medal, the world’s most prestigious mathematics prize. The recipient was Artur Ávila, a researcher at Brazil’s National Institute of Pure & Applied Mathematics (IMPA) and a senior researcher at France’s National Center for Scientific Research (CNRS).

This achievement shows mathematics flourishing in Brazil, as also demonstrated by the First Brazilian Congress of Young Researchers in Pure & Applied Mathematics, held at the University of São Paulo’s Mathematics & Statistics Institute (IME-USP) on December 10–12 with FAPESP’s support.

Ávila, who attended the event at IME-USP, told Agência FAPESP, “People are doing great work, and we’ll certainly hear a lot about them. Mathematics is constantly evolving, and young Brazilian researchers have a great deal to contribute.”

Ávila won the Fields Medal for “profound contributions to dynamical systems theory” that “have changed the face of the field”, according to the prize selection committee. Dynamical systems theory studies systems that evolve over time according to mathematical laws, from the motion of a soccer ball through the air, to the solar system.

“Researchers in this area are trying to say something about systems that are governed by simple laws but in which highly complex things happen in the long run. For example, we learn early on that a planetary system is governed by the law of gravitation. According to this law, a planet orbiting the sun describes an ellipse but considering several planets together in light of Newton’s laws gives behaviors that are hard to describe in time,” said Ávila.

Expressing excitement about the research in his field and other areas of mathematics represented at the São Paulo conference, Ávila went on to say that “what matters most is that there should be no obstacles to the development of research in mathematics”.

Another outstanding mathematician who attended the conference was Pedro da Silva Peixoto, a professor at IME-USP who earned his PhD last year and is preparing to continue his research in the United Kingdom. His doctoral thesis created a theorem that identifies a class of errors in the mathematical models used in weather and climate prediction.

At the University of Exeter, Peixoto will work with an international team of researchers dedicated to developing more accurate mathematical models for weather forecasting. Peixoto leads the research project “Numerical modeling of geophysical fluids on geodesic grids”, which is supported by FAPESP. “Most of the currently used global weather forecasting models have low resolution and will become obsolete in a few years’ time. A good deal of the mathematics behind these models needs to be reformulated and prepared for computational advances in the field. The UK has set up a team of researchers to work on this upgrade, which will make more accurate weather forecasting possible, addressing global phenomena while at the same time predicting conditions in a city neighborhood,” he said.

Peixoto explained that the low resolution to which he referred derives from the limits of existing mathematical models, which use latitude and longitude coordinates to establish coverage areas.

“This results in weather forecasting limited to a resolution of up to ten kilometers, a relatively large area. We intend to enhance the mathematics by dividing this sphere into geodesic grids, which are geometric units such as hexagons and pentagons, affording much richer detail in various parts of the region. However, the mathematics behind this is much more complex,” he said.

Peixoto’s research project will be coordinated by the University of Exeter with collaboration by the Met Office, the UK’s national weather service, in partnership with Imperial College London and other universities. “Brazil wants to enhance its weather forecasting model, and the involvement of Brazilians in international research in this field needs to be encouraged,” he said.

Mathematics without borders

Christina Brech, also affiliated with IME-USP, is working on a different front of mathematics. Her research, which has no immediate practical applications, concerns the interactions between set theory, which studies associations among objects, and Banach spaces. “A Banach space is a mathematical generalization, imitating the environment we live in, for example,” Brech said.

According to Brech, who heads the FAPESP-supported research project “Combinatorial methods in Banach spaces”, set theory can contribute to a better understanding of Banach spaces and help develop the field. In fact, the application of pure mathematics is exactly this, she says: moving mathematics forward.

For Brech the border between pure and applied mathematics has to be crossed. “Mathematics has a very strong aesthetic side, which appears most explicitly in pure mathematics,” she says. “Set theory, which deals with the simplest mathematical objects that exist, with no structure at all, is perhaps one of the areas in which this is most evident.”

At the University of Campinas’s Institute of Mathematics, Statistics & Scientific Computation (IMECC), Olivâine Santana de Queiroz also proposes to solve complex mathematical problems to develop the area on which her work focuses, which is partial differential equations (differential equations in which unknown functions play a role).

“We set out to develop methods and techniques in the area that can be applied to industrial problems relating to the regularity of solutions, for example, but it’s essentially theoretical work and has no immediate interaction with other areas,” she said.

Advances in partial differential equations could ultimately help engineers run more realistic computer simulations of problems faced in manufacturing production processes.

“We’re trying to show that the models used satisfy all the expected properties. That’s the legacy of pure mathematics – enabling knowledge to advance so that its full potential can be leveraged in any application,” Queiroz explained.

New research in mathematics

The conference discussed the distinctions between pure and applied mathematics, as well as other issues for the field such as how best to train new researchers, at a roundtable session with representatives of learned societies and research funding agencies.

“We see great potential for progress in mathematics research,” said Marcos Benevenuto Jardim, a member of FAPESP’s Mathematics & Statistics Committee. “This potential is internationally recognized and needs to be increasingly stimulated, all the way from primary school to higher education.”

During the conference Jardim delivered a presentation on the research funding opportunities offered by FAPESP, including the Young Researchers in Emerging Centers Program. “This is an opportunity to develop potential in a field full of promising researchers,” he said.

Mathematician Jacob Palis, President of the Brazilian Academy of Sciences, researcher and ex-director of IMPA, participated in the discussions and said researchers in the field should be bolder. “This is an important quality for scientists, not just mathematicians. Scientists need to be audacious and may have to break with tradition. Today there are more opportunities, but it’s a mistake to let yourself be imprisoned by them.”

Artur Ávila drew attention to the challenges faced even earlier, by children in high school. “A major obstacle to the development of mathematics in Brazil, despite all the progress achieved to date, is lack of awareness of its potential beyond school, especially in the field of research, which is mostly left in the dark,” he said.

In his view, the way mathematics is presented to children and young people is uninviting. “At an early age, they understand it as something ‘formulaic’, as mere rote learning and rule following, with little room for imagination, whereas it’s the opposite of all that, which is what attracted me to mathematics,” he said.

Ávila believes his winning a Fields Medal could help demystify mathematics for youngsters. “Brazilian research in the area is in evidence, and in a positive way, showing children it exists and is recognized, important. Mathematics is alive and full of enthusiasm,” he said.