15/05/2020

Walner Mendonça defends thesis on Ramsey's theory.

What does a researcher do? At the height of his childhood, Walner Mendonça, from Ceará, didn't have this answer at his fingertips, but he felt that his interests would lead him to the scientific field. "I was very interested in popular science magazines and TV programs about science. However, I wasn't sure what kind of researcher I wanted to be. I imagined I might be a biologist or a physicist." His aptitude for mathematics and his drive to unravel more advanced topics in the subject ended up drawing him to the field.

With the defense of the doctoral thesis scheduled for Monday (18), via videoconference, it can be said that Walner discovered firsthand what the routine of a mathematics researcher is like. Supervised by Professor Robert Morris, the work addresses Ramsey theory in sparse graphs, and its results have connections, for example, in the identification of common interests in groups of users on social networks.

A subfield of combinatorics, Ramsey theory brings together a class of problems with the same paradigm: complete disorder is impossible. The doctoral candidate cites an example that illustrates the problems in this area. “Suppose that, in a social network, all users are friends. However, there are two types of friendships, which we will distinguish by the colors blue and red. Furthermore, every two users share either a red or blue friendship, but not both types of friendship at the same time. This illustration describes what in combinatorics we call a 2-colored complete graph.”

Interestingly, it's not difficult to prove that the social network must necessarily be entirely connected by one of these colors, he points out. "For either of these two colors, let's say blue, it's possible to connect any two users on the social network through a group of users forming a chain of blue friendships." In the figure below, provided by Walner, the illustrated social network has all users connected by the color blue. "Although Letícia and Pedro share a red friendship, we can connect them in blue using, for example, the blue friendship shared between them and Luiz."

Born in Fortaleza, Walner graduated in mathematics from the Federal University of Ceará (UFC). From the start, he felt at home in the course, confirming that he had made the right choice. "In the first semesters, I already realized that I could be happy studying mathematics and that, despite the professional uncertainties in the field, I would continue doing something that I really enjoyed and that was also challenging and even fun."

He began to learn more about modern combinatorics topics through an undergraduate research grant. After completing his undergraduate degree, he decided to continue studying the area in a master's program at the same institution, under the guidance of Professor Fabrício Benevides. “The combinatorics area in the mathematics department at UFC was still very new. Combinatorics is, in fact, a relatively new area in Brazil and doesn't have that many researchers yet,” says the mathematician, who interacted mainly with his advisor. “Fabrício was always a very attentive and patient advisor. It was easy to go to his office and spend hours discussing a problem or an idea.”

It was also through Fabrício that Walner began to approach Ramsey theory, the research area of his then-advisor. “The problems seemed very interesting and appealing,” he says. The variety of techniques needed to study the object is something that intrigues Walner. “The techniques, even for mathematicians from other areas, can seem surprising and ingenious. And indeed they are. But much of the research in the area focuses on improving these techniques in order to obtain a better understanding of the still open problems, which are numerous.” Keeping up with the rapid development of the area in recent years has been one of his greatest difficulties, he says.

The doctoral candidate highlights the support he received from his advisor, Robert Morris, as an important ally in his academic endeavor. “Rob always tried to keep up with the progress of my projects, always being punctual and precise in his observations and suggestions. His mathematical vision is amazing.” Known for attending conferences wearing flip-flops, the British researcher represents for Walner “an example of a humble, simple, and practical mathematician,” which he will cherish for the rest of his life.

Morris's "philosophy of teaching" is something Walner wants to take into his future career as a professor. "In his courses and seminars, he always encourages students to ask questions. As he likes to say, 'there are no stupid questions.' He also likes to let students develop ideas on their own and, as a mentor, always insists that his students frequently come to his office, even if just for a quick chat."

At IMPA since 2016, the Ceará native got to know the institute through the summer program, which he participated in the previous year. Besides being an admirer of the forest surrounding the headquarters, he values the constant conferences in various areas of mathematics held there. “The combinatorics group at IMPA has developed considerably in recent years. We have a large number of students and we have constant visits from various researchers to the institute. Furthermore, Rob's group of students, which is not small, interacts a lot with each other and is always willing to discuss new problems and techniques,” he emphasizes.

Before the institute suspended in-person activities due to the novel coronavirus, Walner was a regular at IMPA's board game group, which met every Wednesday in the coffee lounge. "This was one of the highlights of my week, and I already miss it a lot." During quarantine, he dedicated himself to learning to play the piano and ukulele. "I like to be learning new things all the time," he says.

With his thesis defended, Walner is set to begin a postdoctoral fellowship at the University of Warwick in England, initially remotely. “As soon as the pandemic situation is more normalized, I plan to move to England,” he explains. “Mathematics has given me a way of thinking that I believe few other fields could offer. I even believe it would benefit society as a whole if more people shared this way of thinking that mathematics can provide,” he concludes.