“Esteves founded a school of true mathematics,” says Gatto
“Eduardo Esteves created and founded a true school of mathematics, a mathematical community.” This statement by Italian mathematician Letterio Gatto sums up the tribute to IMPA researcher Eduardo Esteves. Gatto spoke on Monday (31) during the 1Fridays edition of ALGA (Commutative Algebra and Algebraic Geometry), which celebrates Esteves’s 60th birthday.
Titled “Crossing the Boundary, in Search of Limits (Eduardo Esteves: The First Thirty Years of a Mathematical Journey),” the presentation combined the history of science, mathematics, and reflections on the relationship between the two researchers. Gatto explained that he had become familiar with Esteves’s work even before meeting him in person and recalled some of the coincidences that brought their academic paths together.
“When, in early 2026, the organizers asked me to speak about Eduardo’s work, it was a great honor for me,” he said. In retracing this journey, the mathematician pointed out that, although Esteves has worked on different topics—such as curves, linear series, subschemes, modularizations, compactifications, Jacobians, tropical curves, polytopes, and quivers— these topics are connected by a single question: the search for limits.
Gatto highlighted Esteves’s quest to understand the limits of mathematical objects when curves degenerate. The researcher revisited work on the limits of linear series and canonical systems and presented results related to the degeneration of inflection points, known as “flexes.”
The mathematician presented examples of curves that degenerate and discussed what happens to these points and systems during this process. He also presented Esteves’s later work, in which curves can degenerate into more general configurations involving different components and dual graphs. In these cases, residue spaces and polytopes appear in the description of the possible limits.
According to Gatto, although the problems span different periods of Esteves’ career, there is a continuity in the questions he explores. “After 20 or 25 years, Eduardo is still asking the same questions. What is the limit of the canonical system when stable curves degenerate at the boundaries?” he said.
A school of mathematics
In addition to the mathematical results, Gatto devoted part of his presentation to the scientific community that Esteves had built up over the course of his career. The researcher highlighted the network of collaborators and students that had formed around the various topics explored by the Brazilian mathematician.
This insight also caught the attention of younger researchers. Juan Pablo Ocampo, a Ph.D. candidate in Algebraic Geometry and a student of researcher Olivier Martin, highlighted the opportunity to learn about aspects of Esteves’s career that go beyond his scientific achievements.
“I met Eduardo at the beginning of my doctoral program. It was really interesting to get a glimpse into the early stages of his career, see what his work was like, and learn about his personality as well—and, of course, how he built a scientific community. That’s really cool,” he said.
The tribute also moved Esteves himself. “I am very grateful to all of you who have built this community—my colleagues from Brazil and from abroad as well. I am truly very happy; I just have to thank you,” he said.
In closing, Gatto returned to the idea that ran throughout the lecture: the existence of a distinctive style in Esteves’s mathematics. “You can recognize Eduardo without reading his name on the paper, but simply by reading the content of the paper,” he said, comparing this ability to recognize a mathematical style to the ability to recognize Beethoven solely through his music.
“Eduardo has been on a remarkably consistent journey, crossing borders in search of new frontiers. You can still see Eduardo looking toward the horizon, toward a new frontier to cross, new frontiers to discover. And the journey continues,” he concluded.
