11/06/2025

From Harvesting to Geometry: Crislaine Kuster defends her thesis this Thursday.

IMPA doctoral student Crislaine Kuster began learning mathematics the old-fashioned way. In a rural village 14 km from Domingos Martins (ES), Crislaine saw the first seeds of her love for mathematics sprout, planted during practical classes offered by a primary school teacher. Learning the subject took place outdoors through practical, everyday calculations.

Passion blossomed. Daughter of farmers, Crislaine traded harvesting activities in her hometown for Algebraic Geometry in Rio de Janeiro. This Thursday (12), she concludes a four-year journey at IMPA and defends her doctoral thesis “Foliations in homogeneous varieties”. The work was supervised by the Institute's researcher Carolina Araujo. The defense can be followed on IMPA's YouTube channel.

A student at the Eugênio Pinto Santana Public School, Crislaine's interest in mathematics was rewarded with three bronze medals she won at the OBMEP (Brazilian Mathematical Olympiad for Public Schools). These honors held even greater significance in the life of a young woman who divided her time between school and working in the fields. “Back in school, I didn't think much about math. In my routine, I usually went to school and then came back to help my parents. I harvested coffee and bananas. I worked in the harvest with them.”

His academic trajectory began to change with his entry into the Junior Scientific Initiation Program (PIC. Jr.), which he participated in for three years. The next step was his participation in the Hilbert Hotel Meeting in 2014.

“It was the first time I had ever flown on a plane. It was the first time for many things. It was a very fun experience, even though, at that time, I didn't see mathematics as a profession. I was having fun there, learning things that I really enjoyed. OBMEP offered a world and content that I didn't see in the classroom. That really caught my attention,” explained the doctoral student.

The series of activities involving mathematics led Crislaine to graduate from the Federal University of Espírito Santo (UFES), and also to a master's degree in Pure Algebra at the same institution, with a scholarship from the Scientific Initiation and Master's Program (PICME).

In 2019, Crislaine had the opportunity to get to know IMPA through the 32nd Brazilian Mathematics Colloquium. From then on, she became a regular student in the institute's traditional Summer Program, and later applied for a doctorate. "The summer course was a good time, it was really fun, something very new."

Just like during her master's degree at UFES, Crislaine arrived at IMPA to work with Pure Algebra. Her path changed when she fell in love with Algebraic Geometry, the area she works in today. “I managed to get into other places, but at the time, I was very happy to be accepted at IMPA. I had already seen and heard about my advisor, Carolina Araújo, so I was faced with several references that made me think, 'Wow, I want to be there someday.'”

In her thesis, Crislaine works with holomorphic foliations in algebraic varieties. The idea is to study foliations in homogeneous varieties—those that admit the action of a Lie group.

“Foliation is a definition of its variety in subvarieties that, at the intersections of these subvarieties, satisfy a certain compatibility. From this discussion, we try to study what these properties are, how this discussion is done. We also seek to understand if it is possible to extract properties of the variety through the study of these foliations in the variety and so on,” explained the doctoral candidate.

Crislaine points out that her field of work allows for intersection with many areas of mathematics . "Studying foliations "The theory of homogeneous manifolds is very interesting because of this mixture of foliation theory with the theory of homogeneous manifolds. I find it interesting that I'm at an intersection of several areas. There are various techniques you have to study the problem, and different techniques from the same area of geometry, but not so closely related. When they come together, to me, they become very beautiful."

The next step in her career will be a three-year postdoctoral fellowship in Beijing, China, at Tsinghua University, and at Imperial College London, England. During this time, Crislaine will work with Fields Medalist Caucher Birkar, who was awarded the medal at the 2018 ICM, hosted by IMPA.

“I never imagined I would get this far. I think one of the things that makes me very happy about doing research in mathematics is this exchange of experiences. We have contact with the work of people all around the world. Even here in Brazil, we are very involved.” "IMPA , for example, has people from all over and it's a really cool exchange of experiences," said Crislaine.