08/12/2020

Valdir's mathematical journey: from OBMEP to doctorate.

If Valdir José Pereira Júnior had to define his motivations for pursuing a career in mathematics as a sum of factors, he could cite three fundamental points: the challenging problems he solved in his two participations in the Brazilian Mathematics Olympiad for Public Schools (OBMEP), his contact with arithmetic in high school, and reading the book "Fermat's Last Theorem" by Simon Singh. "It's an impressive story that influenced many mathematicians to pursue the field of number theory," says the doctoral candidate at IMPA. On December 14th, he will defend his thesis "Graphs of Hecke operators, orthogonal periods and prime numbers in short intervals," with a live stream on IMPA's YouTube channel starting at 11 am.

“My main research focus is the theory of automorphic forms. Automorphic forms are special functions in certain spaces called locally symmetric spaces. Using an analogy with something classic in mathematics, a bit of poetic license, automorphic forms would be like Platonic solids; they are special objects that possess many symmetries,” explains the doctoral candidate. Valdir points out that when connections between automorphic forms and another mathematical theory are established, these properties of automorphic forms are transferred to the other theory.

With objects, it is possible to associate Galois representations and L-functions. “A fundamental intuition of Langlands in the 1960s was that these Galois representations and L-functions are associated with automorphic forms. Since then, studying this connection has been a fundamental problem in number theory, due to the properties it reveals of Diophantine equations and algebraic number fields. This study is part of what is called the Langlands program.” More specifically, Valdir dedicates himself to two problems concerning automorphic forms, a more recent research topic within the theory.

Valdir's mathematical journey began in rural Minas Gerais, specifically in the city of Paracatu. "I spent my adolescence on a farm, where my interest in studying computer science was sparked. At the end of high school, the OBMEP (Brazilian Mathematical Olympiad for Public Schools) helped me choose a career in mathematics." After finishing his studies at the José Palma Municipal School, Valdir went to Viçosa (MG), where he attended university.

“There, I received an excellent education in mathematics, which helped me a lot when I entered the master's program at IMPA. Academically, my undergraduate studies were when I began studying number theory under the guidance of Professor Abílio Lemos Cardoso. At that time, I studied classical results in algebraic number theory, analytical number theory, and Diophantine equations.” Finally, in 2014, he directed his studies towards algebraic geometry and moved to Rio de Janeiro, where he enrolled at IMPA to pursue his master's degree. “I had heard about the institute through its published books. The institute's distinguishing feature is the wide variety of research areas, where mathematics is lived and breathed, with students exchanging experiences about mathematics all the time.”

While settling into his new city, the doctoral student broadened his mathematical horizons through contact with students and professors of different nationalities and from different regions of Brazil, in seminars and conferences. “I fulfilled my objective of studying algebraic geometry with my advisor, Oliver Lorscheid, with whom I took courses in algebra and algebraic number theory.” Starting from a problem involving automorphic forms proposed by Lorscheid, he arrived at the problem for his thesis and, since then, at the main topic of his academic research.

In 2019, Valdir spent six months in New York, where he worked alongside Gautam Chinta, a professor in the Department of Mathematics at the City University of New York. “We worked on other problems about automorphic forms connected to classical number theory.” Despite the uncertainties of the future due to the coronavirus pandemic, the future PhD in mathematics hopes to close this and other open problems and intends to apply for postdoctoral positions in Brazil and Europe, continuing his trajectory in the world of numbers.