09/09/2019

Omar Alvarez defends thesis in Poisson geometry.

It was the book "What is Mathematics?", by authors Richard Courant and Herbert Robbins, that sparked the interest of Mexican Omar Daniel Alvarez in the field. The son of parents who work in the fields of Philosophy and Literature, Alvarez was captivated by the ideas and concepts presented in the work, more than by the challenges and problems it posed. His admiration for the beauty of Mathematics, which he considers an art form, defined his relationship with the science.

This Tuesday (10), Alvarez takes another step in his professional career. Concluding his doctorate, he will defend the thesis “Integrability of quotients in Poisson and Dirac geometry” at 11 am, in room 236 of IMPA.

Under the guidance of IMPA researcher Henrique Bursztyn, this work addresses symplectic groupoids, which have become a key tool in the study of Poisson structures and are of central importance to the problem of quantization.

For many purposes, it is of fundamental importance to identify concrete classes of Poisson manifolds that are integrable and to have explicit constructions for the associated groupoids. Motivated by these problems, Alvarez focused more specifically on Poisson structures that appear as quotients. As a result, he obtained some new families of symplectic groupoids, among which the moduli spaces of planar bundles on closed and oriented surfaces stand out.

"I think my work addresses a recurring question in the field of Poisson geometry, which is: how to move from classical physics to quantum physics?", he points out.

Brazil was not the first foreign country where the doctoral candidate lived during his 28 years. While still in elementary school, Alvarez moved with his family to Italy, where his father would be conducting new academic research. His contact with art in Turin's museums shaped his relationship with geometry. "Painting and art also carry concepts from mathematics," he states.

Four years later, Alvarez returned to Mexico City to pursue a degree in Mathematics at the National Autonomous University of Mexico (UNAM). From then on, he knew he would follow an academic career. "I didn't consider anything else."

At the end of his master's degree, also completed at UNAM, the mathematician decided to take a summer course at IMPA. The view of the dense Tijuca Forest and the receptiveness of the Brazilians convinced him to stay. He reports that, "Brazil is very similar to Mexico. Both in its qualities and its defects." These aspects made him feel in a familiar environment.

In 2015, Alvarez began his doctoral studies at the institution, where he focused on the area of differential geometry. Beyond personal reasons, such as his appreciation for IMPA's pleasant location, the mathematician highlights that the institution is "a meeting point for high-quality academics." After defending his thesis, Alvarez plans to apply for postdoctoral programs in Brazil.