Renan da Silva Santos presents thesis on Monday (20)

The doctorate at IMPA was a great challenge with a positive outcome in the academic life of Renan da Silva Santos, who presents the thesis “On linked projective spaces” next Monday (20), at 10:30 am, with transmission via IMPA's Youtube channel. After months of not being able to solve problems in his research, the doctoral student changed his approach and arrived at a more intuitive model than the one he initially studied.

Born in Caucaia, in the Metropolitan Region of Fortaleza, Santos said that he was interested in several areas, but, encouraged by teachers at school, he ended up choosing mathematics. During his undergraduate and master's studies at the Federal University of Ceará (UFC), Renan realized that he had made the right choice.

“At an academic level, the doctorate was much more challenging than undergraduate and master's degrees. In general, the mathematics we deal with up to the master's level is already prepared and pre-digested. Working with it is like playing a game. Even if it's extremely complex, you know where you have to get to and what you have to do to solve the main mission. In the doctorate, we actually start doing research. You even know where you intend to go, but the game map is dark and the rules can change. The secret is to persevere until you complete the last mission,” the student compares.

Renan already had an interest in module spaces, but he chose his research topic—limit linear series and representations of special types of quivers—after reading articles suggested by his advisor, Professor Eduardo Esteves.

“Imagine a curve where, on top of each point of that curve, there is a straight line. With some additional properties, this type of structure becomes an invertible bundle. A linear series is a vector space of sections of this structure. If you consider the set of all these sections, we obtain a new curve on top of this set of straight lines. If you deform the original curve into a curve 'broken' into several components, the linear series that was on top of it deforms into a 'broken' linear series, or limit linear series. These objects can be seen as representations of quivers. The objective of my thesis was to study these special representations, called linked networks, and also the space of 1-dimensional sub-representations that are within the larger representation,” details the student, who joined IMPA in 2015, although he already had previous experiences at the institute.

In 2012, he presented a paper at the Scientific Initiation Conference, which earned him a gold medal in the competition. The research was also used as his monograph. In 2014, Santos participated in the IMPA Summer Course, beginning his doctorate the following year, when, after good progress in the initial phase, he encountered some obstacles that needed to be overcome.

“We ended up getting stuck in a dead end for months, without significant progress. Until we finally realized that we were looking at the problem the wrong way and changed our approach. The object of study changed, and that's when the connected projective spaces, mentioned above, came into play. Before, we were working with a similar space, but not as good,” he explains.

When he's not engrossed in mathematical research, one of his hobbies is writing. So much so that the student has short stories, novellas, and even a novel available online. For him, creativity isn't just about words.

“For me, mathematics, like writing, is a way of expressing myself artistically. I see the solution as something very personal and, in a way, unique. If you give the same mathematical problem to two different people, the solution should be essentially the same, but the way of expressing it will not. Each mathematician has their own way of communicating mathematics and adding their personal touches. But, at the same time, these various ways are encoded in the same language pattern, which allows anyone who knows how to read this language to understand,” he comments.