The Olympics shaped the academic trajectory of an IMPA PhD student.

With thought-provoking problems that prioritize logical reasoning over the mechanical repetition of formulas, knowledge olympiads often transform an unassuming student of the discipline into, a few years later, a brilliant professional in the field. This was the case with IMPA doctoral student Cristian Andrés González Riquelme, who defines participation in these activities as a "critical point" in his decision to become a mathematician. "The olympiads allowed me to meet people interested in this universe and to see more closely what it's like to be a mathematician. I began to realize how interesting the problems can be and how fun it is to try to solve them," he says.

A native of Villarrica, a town of 50,000 inhabitants located in the south of Chile, he began participating in mathematics olympiads at the age of 14, while attending high school in the region. From there, he experienced a meteoric rise in this field. He won a total of six medals in competitions such as the Chilean National Mathematics Olympiad, the Southern Cone Mathematics Olympiad, and the Ibero-American Mathematics Olympiad. The high point of his excellent Olympic performance came with a bronze medal at the International Mathematical Olympiad (IMO) in 2012.

On October 5th, Cristian achieved another milestone in his remarkable career. He defended, via videoconference, his thesis "Fine properties of maximal operators," supervised by researcher Emanuel Carneiro.

Research topic in harmonic analysis

Cristian's research focuses on harmonic analysis, a branch of mathematics whose main objective is to understand the oscillatory behavior of certain objects. The doctoral candidate highlights that, within this area, there is a particularly relevant object: maximal operators. "Our research attempts to understand a phenomenon that occurs: since these operators are constructed to control the magnitude of certain things, they are also well-behaved from an oscillatory point of view," he explains.

Cristian learned about the problem through his advisor. The subject attracted him for two specific reasons. “The importance of the type of objects we study and the simplicity of the statements, definitions, and problems, which contrast with the sometimes sophisticated nature of the ideas in proofs,” he lists. Among the difficulties in studying the topic, he highlights the choice of tools. “We work in a context where most classic tools don't work, so we need to build 'tailor-made' tools for these types of problems.”

Experience at IMPA

The Chilean was pursuing an undergraduate degree in engineering at the University of Chile in Santiago when he traveled to Brazil for the IMPA summer course in 2014. “I had heard of IMPA since my first year of undergraduate studies in Chile. I knew that many Chilean mathematicians had done their master's or doctoral degrees at the institute,” he says. After the experience, he was invited by researchers Claudio Landim, deputy director of IMPA, who would become his master's thesis advisor, and Milton Jara, to take a year of short courses at the institution. He left his engineering degree to study applied mathematics at the Federal University of Rio de Janeiro (UFRJ), which he attended concurrently with his master's degree at IMPA.

“Here we have good infrastructure, excellent researchers, and talented students. I was fortunate to have very capable and hardworking colleagues. This is always motivating, and it also helps in case of any doubts,” the doctoral candidate highlighted. In the final stretch before defending his thesis, he laments experiences from this stage of his academic life that were negatively impacted by the pandemic. “I spent a lot of time working alone in my room, which wasn't very healthy. Furthermore, [the pandemic] meant the cancellation of conferences, trips, and all those things that are important for a mathematician in training. But certainly many people were more affected than I was. I can't complain too much,” he shares.

One thing is certain: now that he has his doctorate, Cristian will continue in academia. Among the main aspects he appreciates in a researcher's lifestyle are the freedom to organize his time, the possibility of meeting people from different backgrounds and personalities, and the depth with which he approaches his objects of interest. "I like the fact that researchers dedicate a lot of time to understanding things and overcoming challenges through understanding and creativity, which is very satisfying when things work out," he comments.