Among collaborations, Ana Cristina de Araújo defends doctorate

Ana Cristina de Araújo’s doctoral defense on Monday (12) is the result of an academic career built on dialogue with colleagues, former students and generations of more experienced researchers. Supervised by Marcelo Viana, director-general of IMPA, the thesis continues a line of research developed at the Institute for years, expanding results and incorporating collaborations. The defense of “Hölder regularity of the largest Lyapunov exponent for products of random matrices with a low-dimensional equator” will be broadcast live at 9am by IMPA’s Youtube.

Ana Cristina, 30, from Pernambuco, built her career in mathematics from public school. Born in Recife and raised in the municipality of Itapissuma, in the metropolitan region, she attended elementary and high school at the Colégio de Aplicação, graduated and completed her master’s degree at the Federal University of Pernambuco (UFPE). It was only during her doctorate that mathematics took her out of the state – and directly to IMPA.

Her interest in the area began at an early age and was strongly influenced by scientific olympiads. A gold and bronze medalist in the OBMEP (Brazilian Mathematics Olympiad for Public Schools) and a silver medalist in the Brazilian Mathematics Olympiad (OBM), Ana Cristina took part in the National Awards Ceremony – her first trip to Rio de Janeiro – and in the Scientific Initiation Program, PIC Jr. “I began to see myself as someone who was actually going to study mathematics, because of the Olympiads, because of the PIC”.

Ana Cristina started her degree in Electronic Engineering at UFPE; after three terms, she transferred to a bachelor’s degree in Mathematics and went on to study for a master’s degree at the same institution. It was an assertive decision, she recalls. “I really liked mathematics and thought it was very interesting, it made me want to study. It was a very conscious decision.” Her career was also supported by her family. “Knowing that my family is here to support me has given me the strength to continue firmly on the path I’ve chosen for myself.”

The connection with IMPA began during a summer course. She officially arrived at the Institute in 2022, at an atypical time, right after the pandemic, with part of her doctorate still at a distance. Regarding her routine, Ana Cristina doesn’t hide the fact that she had hoped to have more time to get to know Rio de Janeiro: “I thought I would get out more during my doctorate to enjoy the city, but it’s hard work here, you know? I had to work a lot, a lot, a lot”.

With her regular presence at IMPA, the impact of the academic environment has become evident. For her, the atmosphere of intense work and the constant exchange between students and professors are fundamental hallmarks of training at the Institute. “I’ve never found another place like this, ever. The institute is an ideal place to study and work. The structure here and the conversations with colleagues, teachers and professors push us to continue building and applying our knowledge.” This dynamic of exchange was also directly reflected in the way Ana Cristina learned and researched throughout her doctorate. “My way of learning is very much collectively.”

Ana Cristina’s research is in the area of Dynamic Systems, with a strong dialog with Probability and Ergodic Theory. In general, the work investigates the regularity properties of the so-called Lyapunov exponents, quantities that measure the asymptotic behavior of dynamic systems over time and are fundamental to understanding the stability and sensitivity of these systems to small disturbances.

More specifically, the thesis studies products of invertible random matrices in arbitrary dimensions greater than or equal to two. The main objective is to understand how the largest Lyapunov exponent varies when the probability distributions associated with the system are changed in a controlled manner. The main contribution of the work is to demonstrate that, under certain hypotheses – such as the simplicity of the exponent and restrictions on the dimension of the equator – this exponent presents continuous Hölder-type regularity in relation to a natural metric in the space of probability measures.

One of the highlights of the thesis is precisely its inclusion in a line of research developed collectively at IMPA over the years. The problem studied was divided into three parts, resulting in three theorems, one of which was developed in collaboration with former students of Marcelo Viana who are now working in Costa Rica.

The work continues results previously obtained by El Hadji Yaya Tall, a former IMPA student, who had studied the problem in dimension two. Ana Cristina’s contribution was to extend these results to arbitrary dimensions, advancing an issue already explored at the Institute. Adriana Cristina Sánchez, another former student of Marcelo’s, also collaborated on the research. The group met weekly for months to carry out the research – reinforcing the idea of posterity, collaboration and continuity of scientific production at IMPA.

The process was also one of the most challenging of my doctorate. Working on a technically complex problem, in collaboration with more experienced researchers, required academic maturity and theoretical depth. To master the necessary tools, she studied a recent 100-page article by Marcelo Viana. Despite the difficulties, Ana Cristina highlights the formative impact of the experience. “It was challenging, but it was good. It helped me grow. In order to become a researcher, I have to try to deal with the challenges that arise”.

In addition to her research, Ana Cristina played an active role in initiatives to support women in mathematics during her doctorate. She was part of the organization of events aimed at women in the field and highlights the importance of these spaces. “I think that meetings to promote diversity in mathematics provide a great scientific and personal exchange. I’m very supportive of them and I believe they need to keep happening. I’m completing my doctorate soon, partly thanks to the support of other women mathematicians,” she says, emphasizing the importance of projects such as IMPA’s Olympic Girls (MOI) and the IMPA Women’s Meeting for the training and retention of women in mathematics.

After her defense, Ana Cristina will go on to do a post-doctorate at the Federal University of Ceará, in Fortaleza, maintaining links with the same line of research. In the long term, she intends to remain in Brazil and contribute to the development of mathematics in the country – including strengthening new ties with OBMEP, which marked the beginning of her career.