But, as paradoxical as it may seem, the researchers involved in the mathematical studies that open up all these possibilities are not directly focused on them but rather on developing theoretical models. Due to their sophistication and breadth, they allow us to meet real needs when necessary.
It can be said that researchers even, circumstantially, dedicate themselves to practical demands and they certainly do. However, their biggest concerns are focused on theoretical elaboration because they know that it can contribute to the resolution of real problems, whether closer or remotely.
Theoretical field – Many of these sophisticated studies began with a view to addressing practical situations, but later abstracted them and began to develop in the theoretical field, as the researchers involved like to emphasize. They also remember that theoretical studies, developed and disconnected from any practical problems, allowed the resolution of concrete problems.
These considerations emerge within the scope of the objectives of professors Luiz Antonio Barrera San Martin, coordinator of the thematic project, and Marco Antonio Teixeira, who lead the discussions regarding the research work. Scientists propose the search for new mathematical problems, new lines of research and organization of meetings and events. Researchers Paulo Ruffino and Pedro Catuogno also joined the group. The team also consists of other researchers from Imecc and Unesp in São José do Rio Preto.
About to complete four years, the thematic project headed by professors San Martin and Teixeira, specialists respectively in control systems and dynamic systems, was suggested to Fapesp after an international congress held at Unicamp, organized by members of the group. The most renowned mathematicians in these areas in the world participated in the event. In this line of work, six foreign centers stand out – University of Dijon (France), Autonomous University of Barcelona (Spain), University of Augsburg (Germany), Iowa Satate University (USA), University of Bristol and University of Warwick (England) - in addition to the Brazilian group.
In the researchers' opinion, the developed project fully achieved its proposed objectives, which is reflected in the approximately 60 articles in top quality international circulation journals (with an “A” rating in Qualis/Capes), many in co-operation. authorship with eminent international researchers, and the number of master's degrees (10) and doctorates (15) resulting from it. San Martin admits that the group's comparison with those from central countries is due to the fact that mathematical research does not require many resources. “It fundamentally depends on brains, which facilitates its development in peripheral countries,” he observes.
At the tip – The scientist recognizes that research in mathematics in Brazil is recent, approximately 50 years old, but has already been promoted to group IV (V is the highest) in the international ranking, along with Belgium, Austria and others. “I would like to highlight that in our group Lie semigroup theory is being developed, recognized as fundamental in current research in dynamical systems and control theory. In this regard, we are at the forefront here in Brazil, which is actually not common in other areas of science and technology.”
Regarding the project, Professor Teixeira considers that “we created a framework of what we intended to develop. In the study process, things forked and new questions emerged, in addition to those initially proposed. We managed to resolve most of them and others fell by the wayside, replaced by new demands. The results excite us and give us the right to ask for a renewal of the contract with Fapesp for another four years.”
Teixeira explains that the vast majority of phenomena that undergo changes over time can be modeled by dynamic systems, particularly by differential equations. His group is mainly concerned with the geometric aspects of these modelings, as explicit resolutions cannot be achieved for most of the equations. “So, we study the behavior of solutions without having numerical values.”
Luiz Antônio Barrera San Martin adds: “What we were looking for, in the beginning, was the resolution of these equations. Over time, it was discovered that this was not the best approach to differential equations. More qualitative solutions were then developed. This change in perspective gave rise to the modern theory of dynamical systems. For something that changes over time, we look for a geometric, qualitative solution.”
Paulo Ruffino says that classically solved problems, such as predicting the orbits of the planets, serve as a basis for the work they carry out: “Today, we are concerned with other types of questions. Although our studies are completely abstract, they originated from practical problems. We resolve them and then detach ourselves from them, moving towards a deeper theoretical approach. It is not the objective of our study to link it to the resolution of real problems, although what we do may find situations of immediate application, which does not always occur, because that is not our concern”.
Teixeira recalls, as an example, that the coordinator of a Spanish group with whom he works in cooperation won a competition to determine the most suitable orbit for the satellite used by the European Common Market: “This is a problem that concerns us”. He also recalls that a French group, with which scientists work in cooperation, developed a system that allows maintaining the focus of a camera on a satellite placed over a specific region of the Moon or a planet: “This type of problem is related to dynamic control systems”.
Delight – Ruffino clarifies that the abstract studies they develop have mathematical interest in themselves. “Much of the work of mathematicians is contemplative and results from delight in the results that are being achieved. Researchers savor the beauty of the path of ideas, of following a certain path, the elegance of solutions, like what happens to those who write poetry or texts, or enjoy pictorial creation. It is a beauty compared to the beauty of the arts.”
According to the researcher, from simple situations, the degree of abstraction becomes more sophisticated, reaching a stage where it is difficult to say what the practical application is. “We hope that the results will lead to ever-increasing applications in physics, chemistry, biology, science and technology in general.”
San Martin adds that, if some studies begin with the concern of solving problems, one cannot fail to notice that mathematics later takes on a life of its own, as this is its nature: “It feeds on itself and follows an abstract path. In the future, this knowledge may resolve concrete situations again. That is our hope.”
What a hundred years ago was pure mathematics is now applied mathematics, recalls Professor Teixeira. He clarifies that the objective of pure mathematics is, by eliminating concrete things, to advance in abstractions to arrive at a framework that can be used in the future: “The great power of mathematics is the power of abstraction and not that of calculation”. By the way, San Martin recalls that Group Theory emerged in the 19th century from the need to solve polynomial equations and grew as an abstract theory. “Later, with the emergence of quantum mechanics, Group Theory served as basic mathematics.”
For Pedro Catuogno, the studies they carry out raise some basic questions: how do dynamic phenomena develop over time? What intervening facts matter? In certain circumstances, the conclusions lead to a future time, such as the collision of planets, but there are situations that concern day-to-day life, as in the case of the stock market.
According to Catuogno, the stock exchange constitutes a dynamic system. “Their erratic behavior, which leads people to talk about market volatility and growth rates, can be determined by a stochatic equation, so called because it considers probabilities and random behavior.”
To achieve this, the researcher continues, these equations need to be associated with parameters that allow the insertion of noise that can affect the dynamics of the process behavior. “By 'packaging' all the random factors that constitute noise, the resolution of the equation can be made more realistic and usable by financial institutions, which effectively occurs in banks. The same is done in mathematical modeling that allows weather forecasting.”
For Professor Teixeira, the theoretical contribution of the group should not be forgotten. He recalls that a few years ago the Brazilian Mathematics Council identified four lines of essential research in which Brazil should invest. One of them was Lie Theory (due to Sophus Lie, a 19th century Norwegian mathematician). The group is the only one in Brazil dedicated to this theory, fundamental for the development of dynamic systems.
UA group of researchers from the Mathematics Department of the Institute of Mathematics and Computer Science (Imecc) at Unicamp has been working on a thematic project of international scope. Funded by Fapesp, scientists develop differential equations that study dynamic control systems, so called because they undergo changes over time. In essence, it is a theoretical and abstract study, but one that involves resolving practical issues. 
PRESS ROOM - � 1994-2007 State University of Campinas / Press Office