From Mathematical Olympiads to Research

The path to academic research is both highly personalised and reliant on the learning environment in a student’s family, school, town, and country of residence. Brazilian mathematician Emanuel Carneiro obtained his undergraduate and master’s degrees at the Federal University of Ceará, Brazil, and moved to the University of Texas at Austin for his PhD. He worked at the Institute for Advanced Study in Princeton and the global management consulting firm McKinsey & Company, before securing a position at the Instituto de Matemática Pura e Aplicada (IMPA) back in Brazil. 

Emanuel was introduced to ICTP via his appointment as an ICTP Simon’s Associate in 2016; he became a permanent research scientist in the Centre's Mathematics section in 2018. In the following interview, he talks about his journey from after-school maths classes to working as a mathematician.

How would you describe yourself?

A son, brother, friend, husband, mathematician and father.

I have two brothers. We grew up mostly in Fortaleza in north-eastern Brazil, although we lived in Manaus in the Amazon when I was between four and six years old. When we moved back to Fortaleza, I attended a small school until I was about 10, when I was accepted to study in a military school, one of the best schools in my city.

The teaching was excellent, and that was where I first started to like mathematics. I always liked to do the exercises a bit in advance when my mother bought the mathematics book for the year. I would sit down the summer before classes began, and start to get to know the exercises. And I wasn’t just solving them on scrap paper. Because I knew those exercises would be required as homework, I started solving them very neatly in my notebooks.

What else did you do back then?

I played a lot of video games, and played soccer and basketball as a kid. I liked to go out with friends. I think my view of mathematics became slightly more professional at about 13 years old. There was a Mathematics Olympiad state contest, and students like me who wanted to enter had to do after-school math classes. I entered the competition, and got fourth place.

“I always liked the beauty of math.”

I changed schools again at 15, because I was thinking of seriously pursuing mathematics, and the military school was kind of rigid; they offered extra classes but no more than that. I moved to a school with a great mathematics program, and began to study more and more maths after class.

What did you like about maths?

I always liked the beauty of mathematics. The precision and the truth of a mathematical statement; you can prove it and, once this is done, there's no argument against that. Things are clear. Sometimes you can see there's a proof, or you can prove that something cannot be proved. That's also useful. I like the certainty, when compared to say, human sciences, where they argue their side and you argue your side; there, the truth is more subjective. Also, when choosing a path, you may appreciate certain things, but you also have to be somewhat good at them. I appreciate music, but I don't think I'm good at it. 

I studied much more than what was required for school, and competed in maths contests at the national and international level. I would stay awake nights studying mathematics for six hours on a Friday night. I would go out on Saturday, but on Friday, I’d stay at home and study.

This shows a lot of discipline at a very early age. 

This is probably the biggest lesson that period taught me. If now, somebody tells me I have to work six hours on something and deliver, I can do it. I see how hard it is for teenagers and kids to keep their attention on something now. People are constantly looking at their phones and being distracted. At that age, I was able to think about a mathematical problem for three, four or five hours without stopping.

If you were growing up now, what you think would have been different for you?

Well, I would have more access to information. Despite my hometown being one of the largest in Brazil, I didn't have a lot of material to work with. Brazil was a bit isolated. Nowadays, you can just wander onto the internet and download books; you can buy books online. Then, everything was either unavailable or very expensive. 

I got to a point where I needed more advanced materials than were available in my school. I sent handwritten letters to leaders of mathematics programs in 20 countries in the hopes that some of them would reply. Some did, and with study materials. 

I was able to choose a craft and specialize in it at a very early age. Nowadays, I tell my kids it's good to want to do one or two things well. It could be painting, music, mathematics or science, writing, drawing; whatever you want. This may or may not be something you will do for your life. Some of my friends from the Mathematics Olympiad ended up in medical school. They're probably not using mathematics as much as I am nowadays, but that formation must have helped.

What did you study at university?

After high school, I entered the university to do engineering. At the time, people were seduced by engineering and medical school. I wasn’t so sure I wanted to do science, I didn't even know what science was at the job level, and I decided to do engineering because the curriculum for the first year was the same for various subjects. I did a maths summer class one year, and that was really challenging. I wanted to be challenged at that level.

I still remember telling my father, “So, I think I'm going to change from engineering to mathematics.” He just asked me, “Are you sure you're gonna be able to make a living out of that?” I said I could, and I moved to mathematics, graduated, and continued in the field.

“With a subject like mathematics, you notice that some topics just take you more time than others, and eventually realize that one topic is more natural for you.”

When I began my PhD, I took courses in all subjects within mathematics. When you’re taking these different courses, some subjects strike you as interesting, while others even more so. With a subject like mathematics, you notice that some topics just take you more time than others, and eventually realize that one topic is more natural for you. 

For example, I spent time on algebra, and after a year, I went to a summer school on this and realized I still had a long way to go. I‘d just finished brilliant courses on real and complex analyses, and I’d discuss solutions with my colleagues. They would say, “How did you to solve this problem?” I’d reply, “This idea? Well, this seems to be the only natural idea.” I realized that maybe I was better at that subject.

What do you work on now?

I’d define myself as an analyst in mathematics; a person who likes analysis in its different flavours. More specifically, I'm working on harmonic analysis and some of its applications to number theory. Harmonic analysis is essentially the mathematical study of several types of oscillatory phenomena in nature. This is the pure mathematics that comes at the very, very beginning of the chain of some concepts and phenomena that we have in our real world, such as telecommunications, biomedical equipment, and all types of signal processing and compactification of data.