09/03/2021

Vasquez defends thesis on the bispectral problem.

If it weren't for mathematics, Brian Vasquez might still be working in a clothing factory in Colombia, a job he held between high school and his dream of entering university in 2009. “It was a very difficult period, but entering the field of mathematics changed my life. I achieved many things because of my studies. I owe everything I have achieved to mathematics,” says the future PhD from IMPA. Next Wednesday (10), the Colombian will defend his thesis “On the Matrix Bi-Spectral Problem”, starting at 2 pm, with transmission on IMPA's YouTube channel.

In the work supervised by Professor Jorge Zubelli , Vasquez analyzes algebraic properties in a general framework and derives a method to verify whether a polynomial matrix subalgebra is bispectral. Furthermore, he performs an analysis, in terms of generators and relations, comparing it to some finitely presented algebras. “In the first example, we present a Pierce decomposition of this algebra. As a byproduct, we positively answer a conjecture by F.A. Grünbaum regarding certain non-commutative matrix algebras associated with the bispectral problem. And we also prove matrix bispectrality with polynomial potentials that satisfy a second-order autonomous matrix differential equation.”

The connection with the research topic began during his master's degree, also at IMPA. “I was supervised by Professor Roberto Imbuzeiro, and our focus was on harmonic analysis. But Professor Zubelli proposed that I solve the bispectral problem,” he recounts. The partnership yielded good results: in addition to supervision during his doctorate, after the defense, Vasquez will be a research associate at Khalifa University in the United Arab Emirates, where Zubelli currently teaches. The two also intend to publish the results recorded in the thesis in academic journals.

Born in Ibagué, a peripheral city and capital of the Tolima department in Colombia, Vasquez inaugurated a new space for his family. “I am the first of this generation to enter university and pursue a career in academia,” he says proudly. His journey in higher education began in 2010 when he enrolled at the University of Tolima. “At the time, I wanted to study medicine, but due to the results of the Colombian university entrance exam, the ICFES (Colombian Institute for the Evaluation of Education), I wasn't accepted. Mathematics was my second choice, so I ended up pursuing it.”

A few months later, more precisely in the third round of reclassification at ICFES, the then-undergraduate mathematics student was approved to begin his medical career. "But I understood that it was too late, I was already immersed in fundamental mathematics and I didn't want to leave anymore."

The end of her undergraduate studies coincided with her arrival in Brazil for her first Summer Course at IMPA in 2014. Seeing references in the classroom that she had encountered in books back in Tolima, she had no more doubts. The academic path was the best option to follow. “Besides IMPA being a leading institution worldwide, it’s a very well-organized institute, unlike my experience in Colombia. That’s why I decided to stay.” To do so, she moved to the Horto neighborhood in Rio.

“Between my master’s and doctoral studies, I stayed in several houses in the region. The first was the one with the brown door, and the last was the one with the red entrance. They are quite famous residences among the students,” he recalls fondly of the region, since it was from Colombia that he took the final steps towards completing his doctorate, due to the coronavirus pandemic. “It was very important to be with my family during this turbulent time. And returning here showed me how important education is. I intend to continue on this path,” he affirms.