Caio Melo defends his doctoral thesis on veneers on Wednesday (8)

Second choice in the entrance exam or, at most, an alternative project after the much-desired engineering course, this was the space that mathematics occupied in Caio Melo’s plans. The scenario changed when he took his first class for a bachelor’s degree in mathematics at the Federal University of Rio de Janeiro (UFRJ). “I realized it was what I wanted to do, it enchanted me, that’s how I fell in love with mathematics,” he recalls.

This Wednesday (8), Melo defends his doctoral thesis at IMPA, at 10:30 a.m., in room 232. Supervised by IMPA’s deputy director, Jorge Vitório Pereira, the work presents advances in the area of foliations.

While still an undergraduate, he had his first contact with IMPA through the Summer School. The experience was decisive for him to later apply for a master’s degree at the institution. Without great expectations, he was approved and began his studies in a challenging context: the beginning of the Covid-19 pandemic. “IMPA managed to adapt well to remote teaching, without any major delays,” he says.

Happy with his choice, he continued at the Institute for his doctorate. “I came to IMPA with high expectations and I think they were met. The professors are very good. I’ve learned a lot, both mathematically and about life,” he says.

One of the highlights of her career was her international experience during her sandwich doctorate at the Université de Rennes in France. The period was remarkable both academically and personally, broadening his horizons and strengthening his education.

His thesis, entitled “Transversally projective foliations of codimension one”, is in the field of pure mathematics and investigates structures known as foliations. Simply put, these structures can be understood as ways of decomposing complex spaces into “layers”, similar to the pages of a book.

The work focuses on a specific type of these structures, analyzing both regular cases and situations with singularities – points where the behavior changes, like eddies in a stream of water. The research is organized on two main fronts.

The first deals with the classification of certain foliations on special geometric surfaces, called compact Kähler surfaces. The second part was inspired by the so-called Touzet Conjecture, an open problem in the area, and seeks to understand when these structures can be described in a simpler way.

“One of the ideas is to see when we can reduce something very complex to a simpler function,” explains Melo. At this stage, he obtained results on conditions that guarantee the so-called algebraic integrability of these foliations.

Part of the research was carried out in collaboration with mathematician Gabriel Fazoli, who holds a master’s and PhD from IMPAbased on a research proposal suggested by his supervisor.

Having completed his doctorate, Melo is looking to the future. Recently approved for the Rio de Janeiro State University (Uerj), he is waiting to be called up, without losing sight of his interest in academic research and post-doctoral studies.