23/08/2018

Adriana Chavarría defends the passion for "chaos"

As a young girl, Adriana Sánchez Chavarría, from Costa Rica, realized she loved mathematics. In high school classes in the capital, San José, learning with numbers was more than just a school assignment; it was a pleasure. Her fascination with the subject was so great that she even shared what she had learned with her classmates. When it came time to choose a career, she saw that she couldn't escape mathematics.

At 4pm this Friday (24), in room 232, Adriana will defend her doctoral thesis at IMPA. Supervised by the general director, Marcelo Viana, the thesis “Contributions to the continuity problem for Lyapunov exponents” will be presented in room 232.

"In my search for a profession related to numbers, I found Actuarial Science, a career at the School of Mathematics at the University of Costa Rica (UCR) related to economics and risk analysis," he says.

Between 2004 and 2006, Adriana studied the same subjects as in the Mathematics course. The more she learned, the more she enjoyed studying and researching mathematics. It didn't take long for her to realize she was in the wrong course. "I changed fields and decided to dedicate myself to Mathematics."

Passion for chaos

The thesis crowns Adriana's passion for Dynamical Systems, acquired during her undergraduate studies in the classes of the only dynamical system instructor in Costa Rica, Professor William Alvarado, from UCR. "I took a course with Alvarado on chaos theory and fell in love with dynamics."

After graduating, the researcher began teaching in the Engineering and Economics courses at the University of Costa Rica. She also completed a degree in Actuarial Science and decided to try for a postgraduate position at an institution abroad. UCR does not offer a doctoral program.

In 2011, he came to study at IMPA, where he completed his master's degree under the guidance of Robert Morris. Now, he is completing his doctorate under the supervision of Marcelo Viana.

Inside the thesis

The doctoral thesis in the area of Dynamical Systems and Ergodic Theory focuses on the continuity of Lyapunov exponents in two cases: partially hyperbolic diffeomorphisms and non-compact support measures.

For those unfamiliar with the process, Adriana explains how it works. “Imagine you have matrices A_1 and A_2 and a bag with two balls, one white and one black. In the first round, you take a ball from the bag. If it's white, you choose matrix A_1. If not, A_2. And you return the ball to the bag. In the second round, you do the same and multiply the matrices. We repeat the process several times.”

But here's the catch… imagine that, instead of having two matrices, there are an infinite number of them. Thought that was all? It can get a little more complicated: the bag has an infinite number of balls, all with different numbers.

“The result of the process is a random product of infinitely many matrices. The Lyapunov exponents control the growth of the product norm. I'm trying to understand, for example, what happens if I replace one of these matrices with another very close to it. Do the exponents also become close?” he asks. The answer? It's in the thesis!

Second home

Adriana says that, initially, Viana wasn't her advisor. "Even so, he took some time to try and calm my nerves before the qualifying exam, always telling some story from his past."

During the three years they worked together, the researcher remembers him as a very understanding professor. "He manages to explain complicated things in a very simple way," she says.

According to the Costa Rican, IMPA has exceptional researchers. "So, it's always possible to find someone to talk to and ask questions."

As soon as she is approved, Adriana will pack her bags. She will leave Rio de Janeiro and, consequently, IMPA, her "home" for seven years, to live in São Carlos (in the interior of São Paulo), where she has been accepted as a postdoctoral researcher at ICMC-USP, under the supervision of mathematician Ali Tahzibi.

SERVICE:

Thesis defense by Adriana Sánchez Chavarría

“Contributions to the continuity problem for Lyapunov exponents”

Date: August 2Wednesdays | Time: 4 PM | Location: Room 232