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From Materials to Medical Imaging, Fonseca’s Work Shapes the Future of Innovation

ºÃÉ«ÏÈÉúTV’s Irene Fonseca uses the power of math to make everyday life better. Her ideas help create stronger materials, shaper images and new tools in engineering and physics, showing that even the most abstract math can spark real-world breakthroughs.

The Kavčić-Moura University Professor of Mathematics and director of CMU’s Center for Nonlinear Analysis (CNA) works at the crossroads of mathematical theory and practical problem solving. Her research spans two influential areas: how advanced materials behave and the mathematics behind computer vision.

Using mathematics to improve materials and images

In materials science, Fonseca develops mathematical tools that help physicists and engineers understand how novel materials bend, break, transition and respond to external forces. These insights guide the creation of technologies ranging from minimally invasive surgical devices to sustainable energy systems.

“That’s the beauty of math. It’s this abstraction where you’re not wedded to concepts or exact formulations,” Fonseca said. “I get a motivation from applications, and I bring them into the math context to try to formalize them.”

Her work provides a rigorous framework that helps researchers refine materials before they are manufactured, which saves time, resources and energy while opening the door to breakthrough collaborations.

In computer vision, Fonseca’s expertise supports everything from medical imaging to cultural preservation.

These problems, although very different on the surface, are held together by a common mathematical thread.

“In materials, if there is a deformation, there is a blob that you need to deform or in a sharp area, it might be a fracture in the material. In imaging, if there’s a blob, it could be a face and the sharper area could be the edge of a feature,” Fonseca said. “If you don’t put labels, the mathematical models and questions you ask are basically the same. And the tools are basically the same albeit formulated in different languages. And the mathematical variational tools are very similar.”

Solving problems too complex for traditional tools

Many physical and technological systems behave in ways that exceed the capabilities of classical mathematics. Defects in materials, rapid changes known as sharp transitions, multiscale behavior and competing forces all contribute to systems that defy standard models. Similar challenges arise in image processing, where computers must recognize edges, shapes and patterns despite noise or missing information.

Fonseca’s work advances the mathematical foundation needed to tackle these complexities. By developing new tools that account for abrupt changes, higher-order effects and interactions across scales, she helps scientists make sense of problems once considered too difficult to analyze.

Her contributions fuel advances in next generation medical devices, flexible electronics, advanced alloys and energy efficient technologies.

Where math meets machine learning

Materials science and computer vision share structural similarities. With this in mind, Fonseca built a toolkit that blends classical analytical mathematical techniques with modern machine-learning methods that can be used across both/many fields. This integrated approach is helping researchers push beyond traditional limits and uncover new ways to model and interpret complex systems.

At Carnegie Mellon, Fonseca serves as a co-principal investigator on a National Science Foundation (NSF) new research training group (RTG) in applied analysis. The initiative brings together researchers across colleges and international partners to apply sophisticated mathematical models to solve next-generation challenges in science and technology.

Through the RTG grant, the CNA will be launching the Mathematics Outside of Mathematics (MOMA) seminar later this year. MOMA will be a monthly series featuring speakers from industry, national labs and applied academic fields who demonstrate how mathematics drives innovation.

A global leader in applied mathematics

Fonseca’s influence extends beyond her research. She is a fellow of the American Mathematical Society, the European Academy of Sciences and the Society for Industrial and Applied Mathematics (SIAM). Her honors include a knighthood from the President of Portugal, and the Senior Prize from the International Society for the Interaction of Mechanics and Mathematics — awarded for both breakthrough research and exceptional mentorship.

As a past president of SIAM — the world’s largest organization devoted to applied mathematics and computational science — she helped shape how the mathematics community engages with industry, government and rapidly evolving technology sectors. She continues that leadership as a vice president of the American Mathematical Society.

Fonseca sees training young researchers as central to her mission. A faculty member at Carnegie Mellon since 1987, she has guided students and early career researchers whose work now spans multiple scientific domains. Many of her trainees also work at the intersection between materials science and computer vision.

“It is paramount having graduate students and postdocs. It’s more important to me than publishing, being able to train them and help them on their paths of becoming even more excellent researchers,” she said. “It’s very rewarding to me now that I see them as leaders in their own rights and having their own groups.”

For her efforts, she has been elected a fellow of the American Association of the Advancement of Science (AAAS), the world’s largest general scientific society and publisher of the Science family of journals. The fellow distinction is a lifetime honor achievement within the organization, which promotes the use of science, technology, engineering and mathematics to solve current issues.

Earlier in March, she also was announced as the latest recipient of the University of Lisboa Prize (Prémio Universidade de Lisboa / Caixa Geral de Depósitos), which was awarded for her role in the international mathematical community and her enduring contributions to research, education and scientific inquiry.