Seminar Explores Connections Between Algebraic Geometry and Cryptography

How does a branch of mathematics that has evolved over centuries help protect information in the digital age? That was the question that guided this Tuesday’s (16) IMPA Tech academic seminar, led by IMPA researcher Carolina Araujo. Under the theme “Curves, Equations, and Secrets: A Journey Through Algebraic Geometry,” the event brought together students in the Bachelor’s program in Mathematics at the School of Technology and Innovation for an introduction to one of the most traditional areas of mathematics and its contemporary applications.

The presentation began with a historical overview of Algebraic Geometry, a field that combines concepts from geometry and algebra to study solutions to systems of polynomial equations and their geometric interpretations. Throughout the lecture, Carolina demonstrated how these two perspectives complement each other in understanding complex mathematical structures.

“It’s a pleasure to be here to talk about my field of research and share an application. The line between pure and applied mathematics is a fine one. I want to show you how we can apply these concepts to cryptography,” he said.

According to the researcher, Algebraic Geometry occupies a unique place in the history of mathematics. “It’s a very old field, situated at the intersection of geometry and algebra,” she explained. During the presentation, she highlighted the importance of the introduction of Cartesian coordinates for the development of the field. “For each point on the plane, we have x and y coordinates, which generate an equation associated with each point. Algebraic geometry studies solutions to systems of polynomial equations in several variables. These solutions have a geometric manifestation. We build bridges between these two perspectives,” she said.

In the second part of the discussion, Carolina presented applications of algebraic geometry in cryptography, the field responsible for protecting digital information and communications. The researcher recalled historical examples of message encryption, such as the methods used in Ancient Rome, and explained the principles underlying modern cryptographic systems.

“Think about it: we have two characters who want to communicate, but in a way that no one who intercepts the message can understand its content,” she suggested to the students. Building on this idea, she introduced concepts related to public-key cryptography and the Diffie-Hellman method, which is used for the secure exchange of information in digital environments.

The lecture underscored the importance of a mathematical education for understanding the technologies we encounter in our daily lives. By linking abstract concepts to concrete applications, the seminar highlighted how fields traditionally associated with pure mathematics can contribute to addressing contemporary challenges in computing and information security.

A researcher at IMPA since 2006, Carolina Araujo holds a Ph.D. from Princeton University and works in the field of algebraic geometry, focusing on birational geometry, Fano varieties, and algebraic foliations. The first Brazilian to receive the Ramanujan Prize in 2020, she was also awarded the L’Oréal–UNESCO For Women in Science Prize in Brazil. In addition to her scientific work, she has played a prominent role in promoting women’s participation in mathematics and served as chair of the International Mathematical Union’s (IMU) Committee on Women in Mathematics from 2023 to 2026.

For student Paula Pires, from the class of 2026, the seminar was an opportunity to explore a field she had not yet delved into. “It was one of the best seminars I’ve ever attended. The researcher was an excellent teacher, and her research topic is very interesting. It was especially intriguing to have this first—albeit introductory—exposure to concepts in algebraic geometry. The connection drawn between the content and cryptographic methods also inspired me to research the subject further,” she said.

Paula also highlighted the importance of female representation in science. “It’s motivating to see that other women have already reached the positions I’d like to hold one day. Today, I have the privilege of spending time every day with women who inspire me and reinforce my belief that this path is possible,” she said.

For the student, the seminars and outreach activities organized by IMPA Tech expand educational opportunities beyond the classroom. “The opportunity to attend a variety of seminars led by prominent researchers and to learn about topics that aren’t typically part of the traditional curriculum is a great privilege. These experiences foster extraordinary personal and professional growth,” she concluded.