20/10/2022

Fields Medalist Smirnov discussed percolation at IMPA 70.

Complex analysis, dynamical systems and probability theory filled Stanislav Smirnov's enthusiastic lecture during the IMPA 70th Anniversary Conference this Thursday (20). Winner of the Fields Medal in 2010, the highest honor in mathematics, the Russian delighted the audience with 2D percolation drawings that, as the mathematician joked, could have come from "children's coloring books".

Despite the lighthearted tone, the topic presented is no child's play. In an hour-long presentation, the researcher from the University of Geneva showed various applications and challenging questions about the study that intertwines geometry and chance. The task is not simple, but Smirnov considers it enjoyable. “If you study the connections, you can create interesting patterns! For example, if you cut the city in two, there is always a connection between the right side and the left side,” the researcher explained.

Roberto Viveiros, who graduated with a doctorate from IMPA in 2020, attended the lecture by the Russian mathematician. “The first article I read on probability was by him, so I'm somewhat familiar with some of the things he said, but not with others yet. I really like the topic! Within the field of probability, percolation is one of the most beautiful,” commented the IMPA PhD.

IMPA researchers present geometry from a different angle.

Another lecture presented this Thursday was by IMPA researcher Reimundo Heluani, who spoke about factorization. The mathematician discussed the relationship between representation theory and algebraic geometry theory. “You have a symmetry structure that is completely abstract and you want to represent this structure acting in some concrete place. One example could be the symmetries of the Rubik's Cube, which are a completely abstract matter,” he explained.

Earlier, IMPA researcher Lucas Ambrozio presented to the public the results of a study that starts from an old problem, analyzed by the mathematician and physicist Eugenio Beltrami, back in the 18th century. At the time, the scholar was analyzing the possibility of drawing maps of the Earth's surface such that the shortest paths between two cities are represented by straight line segments.

Initially, the mathematician's goal was to obtain information on the shortest routes between two points. However, he discovered that such maps only exist under a special circumstance: the Euclidean sphere. Using this model, Ambrozio expanded the analyzed dimensions and obtained information that Beltrami had never imagined.

“Instead of talking about a two-dimensional sphere, we talk about a sphere of any dimension, N; instead of curves, we talk about minimal hypersurfaces, and from this, we see new things that Beltrami hadn't imagined: there are examples that have no symmetry, no particular constant curvature,” he explained.

Patrícia Gonçalves spoke about particle systems.

Also present at the “IMPA 70th Anniversary Conference” was researcher Patrícia Gonçalves, from the Instituto Superior Técnico (IST) in Lisbon. She presented to the participants a study that seeks to understand the movement of gases in a space based on probability.

“The idea is to try to deduce the laws that govern the gas or other physical system and understand how the gas spreads. The way to approach the problem is to assume that the molecules move randomly and, using this rule, try to discover what I will observe at a certain point in time,” the researcher highlighted.

Before beginning her lecture, Gonçalves mentioned that she was a student of researcher Claudio Landim, deputy director of IMPA, in a summer course in 2002, and spoke fondly of her relationship with the institute.

“I experienced a culture shock here, and that was very good. IMPA has a friendly teaching environment. I arrive here and receive many hugs, and that's unique. We enter and our hearts are full,” she said fondly.

: Reimundo Heluani (IMPA)

: Patrícia Gonçalves (IST, Lisbon)

: Lucas Ambrozio (IMPA)

: Stanislav Smirnov (Genève)