From Whys to Turbulence: Erika Bernal defends her thesis at IMPA

From an early age, IMPA doctoral student Erika Bernal learned that the world came with ready-made answers. It was in her teens that the turning point came: questioning. Why? Why do phenomena happen? Why are things the way they are? Amid doubts about nature, life and human behavior, mathematics appeared as the way to transform curiosity into scientific investigation.

Continuing in this vein, this Friday (6), Erika is defending her thesis “Spontaneous Stochasticity and the Inviscid Limit in Incompressible Flows in Logarithmic Networks”, supervised by IMPA researcher Alexei Mailybaev. The defense will take place in room 232, at 10am, and will be broadcast on IMPA’s YouTube channel.

From Tame, Arauca, Erika moved to Paz de Ariporo, in the department of Casanare, Colombia, at an early age and was the first in her family to finish university. Before studying mathematics at the National University of Colombia in Bogotá, science seemed far away. “The only contact I had was through television, but it was so far away from me. I thought it was very difficult, very bright people were doing it. So math was never an option for me.”

Throughout his degree, he discovered not only the pleasure of mathematical demonstration, but also the beauty of applications. “I always liked to see how mathematics applied to the real world, to nature. You can explain things in nature, things that happen, through mathematics.” The possibility of pursuing research brought her to Brazil, first for her master’s degree and then for her doctorate at IMPA. Her arrival coincided with the start of the pandemic, in 2020, and was accompanied by the challenges of remote teaching. It was also the beginning of a deeper academic career for the young woman.

During her doctorate, Erika dedicated herself to one of the most challenging topics in applied mathematics: turbulence. Present in everyday flows and astrophysical plasmas, it is described by equations famous for their complexity, such as the Navier-Stokes equations, whose complete understanding remains open. Issues such as the formation of singularities in finite time (the so-called blow-up), spontaneous stochasticity and the behavior of solutions in the inviscid limit are among the major problems in this area.

In her thesis, Erika investigated these phenomena using logarithmic network models, an approach that reorganizes the space in which the equations are analyzed. The technique preserves essential properties of the original equations, but reduces the computational cost, making it possible to numerically explore extreme regimes that would be inaccessible directly.

With this strategy, she presented evidence that chaotic blow-up is a robust phenomenon and observed, in three dimensions, signs of spontaneous stochasticity in different scenarios. One of the points she highlights as most important in her work is precisely that she obtained numerical evidence of this phenomenon in a model that is closer to the original equations than the simplified versions previously studied. The results reinforce the importance of logarithmic networks as an effective tool for investigating extreme dynamic regimes in fluid mechanics.

When he finishes his doctorate, the feeling is ambiguous: “It’s a mixture of something good and also melancholy. Good for being able to finish, to complete this part. But it’s also sad, because we’re leaving, we’re leaving our friends.” Still, gratitude prevails. “Overall, I feel grateful for everything I’ve experienced here.”

Erika would like to stay in Brazil and continue her research, making mathematics not just a profession, but a way of questioning and investigating. In Rio de Janeiro, she found similarities with Colombia. “I really like it here and I’d like to stay. I like the ‘noise’ of the city. The people are friendly and very cheerful.”