16/02/2023

For Ambrozio, mathematics is a 'creative exercise'.

Ana Luísa Vasconcellos

Through the lens of 34-year-old Lucas Ambrozio, the world can be interpreted from the geometric properties of curves and surfaces. Shy and always with a friendly smile on his face, the IMPA researcher walks calmly through the corridors of the institute where he earned his doctorate, usually dressed in a light blue dress shirt and jeans. His research, focused on differential geometry and geometric analysis, demands a great deal of knowledge of the discipline and a touch of imagination from the mathematician.

“I see mathematics as a creative exercise. Doing mathematics is very similar to any creative process, whether in music, literature, or other forms of expression. It's about seeking to create something different, but following a set of rules. Just as a poet needs to obey certain criteria to write a sonnet, such as respecting the number of stanzas and the rhyme scheme, a mathematician needs to organize their thoughts to create their theorems, respecting rules of argumentation. And yet, there are so many possibilities!” said Ambrozio.

A researcher at the institute since October 2020, he is primarily interested in minimal surfaces, that is, those made of small pieces, each with the smallest possible area among the surfaces that delimit the same boundary. While the study of geometry, whose basic concepts are introduced in school, may seem simple to laypeople, for Ambrozio it is an inexhaustible field of possibilities.

“Like all students of geometry since Euclid, we are interested in simple geometric concepts, such as the length of curves, the area of surfaces, and curvature, but we analyze objects using tools from differential and integral calculus, and partial differential equations,” he explained enthusiastically.

His connection to curves is also present in his childhood. Born in Rio de Janeiro, he moved to Petrópolis when he was a few months old and, at the age of three, to Teresópolis, after the birth of his brother. Surrounded by the tranquility of the mountains of the Rio de Janeiro highlands, he discovered his love for learning early on.

“I wanted to learn everything! I’ve always had a liking for mathematics and perhaps an aptitude for it. But it only emerged as a future possibility for me much later,” recalled Ambrozio, who considers himself more of a mountain person than a beach person.

He grew up in an environment that valued education. His mother, a speech therapist, was an early childhood and special education teacher for many years and is now retired. His father graduated in architecture and now works in computer programming, a career that the youngest member of the family also decided to pursue.

“In my home, everyone has always done their best to foster this curiosity and support our studies. They don't fully understand what I research, but I know they understand what a mathematician does, which is rare, and an important support,” he reflected.

Passion for the 'language of the universe' ranges from literature to mathematics.

Ambrozio recalls being impressed during his adolescence by the studies of Einstein and other physicists, who viewed the "nature of the universe" as a geometric form that evolves over time. At the same time, he did not rule out the possibility of becoming a writer, as he devoured classics of Brazilian and foreign literature.

"Since I was an avid reader, I even thought about writing. But nowadays, I think that was more a result of being a reader than of having a story to tell," he recalled humorously.

Considered by the most enthusiastic researchers as a "language of the universe," mathematics ended up emerging as a professional choice in high school. One of his teachers, Marcela Amorim, had participated in the High School Teacher Improvement Program (PAPMEM), offered by IMPA. She told the young man that, in Rio de Janeiro, there was a place where researchers explored this language.

Curiosity to explore this "special" place, surrounded by the greenery of the Botanical Garden and inhabited by great mathematicians, led Ambrozio to enroll in IMPA's Summer Program in 2006. He had the opportunity to take classes with researcher Elon Lages Lima, former director of the institute, considered one of the leading figures in Brazilian mathematics, who passed away in 2017.

When she finished the course, she was certain she was in the right place. “Mathematics is very creative and, therefore, needs an environment conducive to that creation! It is diverse and needs to remain diverse, because everyone has their own way of doing mathematics, so it’s important that there are spaces for that,” she reflected.

He graduated in mathematics from the Federal University of Rio de Janeiro (UFRJ), where he was mentored in his scientific initiation by Professor Walcy Santos, and returned to IMPA at the first opportunity.

“When I came to IMPA, it was in a hurry. In the middle of my undergraduate studies, I already wanted to be here. And I started my master's degree in my last year of undergraduate studies. The decision to pursue the field of geometry came in my second year, when I took Fernando Codá's differential geometry course. I was fascinated by the things he was researching,” he said.

During his doctoral studies at IMPA, he was supervised by Codá, who proved Willmore's Conjecture with André Neves in 2012. The problem is famous in differential geometry and had remained unsolved since 1965.

“Codá is, without a doubt, one of my greatest inspirations. He greatly influenced my understanding of what the important questions in geometry are, and how to present them in classes and lectures,” said Ambrozio.

He defended his thesis in 2014. To complete his impressive resume, he did postdoctoral work at Imperial College London and the University of Warwick, also in England. And he got to know firsthand the place that had physicist Albert Einstein as one of its pioneers, the Institute for Advanced Study (IAS) in Princeton. Ambrozio was a member of the IAS between 2018 and 2019.

Creativity beyond scientific production

For Ambrozio, the research process is long, enjoyable, but often frustrating. “Mathematics has many points where you end up banging your head against the wall because you can't move forward anymore. But when you manage to organize your thoughts to prove your reasoning and write down your conclusions, it's very rewarding. It's one of the best parts of the process,” he commented.

And when they can't find the answers in mathematics, music and literature serve as an escape.

“My main leisure activities are reading and playing the violin. Lately, I've been reading more non-fiction and things that are closer to lived history, like biographies. Right now, for example, I'm reading the book by Davi Kopenawa, a Yanomami shaman,” he stated.