26/07/2023

'It's an honor to return to CBM as a speaker,' says Laura.

Researcher from Harvard University (USA), Laura DeMarco opened the lecture “ Rigidity, uniformity, and the Mandelbrot set ” , on the afternoon of this Wednesday (26), recalling her visit to the Brazilian Mathematics Colloquium of 1999 – when her advisor was part of the event's panel of speakers. “It is an honor to return as a speaker”, said the researcher, who addressed the relationship of the Mandelbrot Set with other areas, including algebra and number theory.

The Director-General of IMPA, Marcelo Viana, was responsible for introducing the speaker. “Laura graduated from Harvard and has become a world reference in Complex Dynamics and Arithmetic Dynamics. She is a member of the US National Academy of Sciences and was a speaker at the International Congress of Mathematicians (ICM) here in Rio de Janeiro in 2018.”

“The Mandelbrot Ensemble plays a fundamental role in current research, so in this lecture I will explain its importance and why it has been studied for over 40 years, and finally, we will see some examples of how we are working with it,” Laura said at the opening of the lecture.

In an energetic manner, the mathematician showed images of fractals and presented some points from her research.

For IMPA doctoral student Miguel Ratis, who studies the analytical part of the Mandelbrot Set, it is not yet possible to predict whether Laura's results will one day be useful for his research. “We're talking about mathematics. So, you never know when you'll stumble upon a problem and say, 'whoa, now I need this information that Laura is working on.' These are very new results; it's still difficult to know how influential they might be for the type of research I'm developing with Luna (Lomonaco, IMPA researcher).”

José Victor da Trindade Medeiros, also a doctoral student of Luna's, highlighted Laura's characterization of Mandelbrot as a place of universal chaos. "In several holomorphic families that appear in dynamical systems, when you drastically change the dynamics, Mandelbrot appears, which is a very good thing."

Camillo De Lellis

Researcher at the Institute for Advanced Study (USA), Camillo De Lellis opened the first CBM lecture this Wednesday (26). With the title “Area-minimizing integral currents: singularities and structure” , the researcher cited the book Differential Geometry of Curves and Surfaces, by the mathematician and IMPA researcher, Manfredo do Carmo (1928 – 2018), as responsible for his first positive contact with the subject.

“In my first year of study, my classmates and I got very lost in these concepts, until one of us said: 'well, there is this wonderful book [Differential Geometry of Curves and Surfaces] by Manfredo do Carmo that we could read' and as soon as we opened the book, we began to understand everything. It was the first time I encountered this notion of minimal surfaces,” he told the audience.

De Lellis also stated that "it is a great pleasure to be at the colloquium given IMPA's history," especially in this area.

Lellis' student, Reinaldo Rezende, who recently completed his doctorate at Princeton University, explains that the Theory of Minimal Surfaces is a very famous topic in mathematics.

“Minimization problems come from physics, from nature. For example: you have a mountain and you want to build a road and you need to minimize expenses, costs, deforestation. That's a type of minimization problem. But there are also singularities, which are obstacles to taking abstract mathematics into an applied context. So, we want to get rid of them, and Camilo's main theorems show that singularities exist, but they are few in the mathematical sense,” explained Rezende.