Mathematical Imaging in the Era of AI
Mathematical imaging has for decades been a powerful catalyst for new developments across the mathematical sciences. At its core lie fundamental questions about how information can be represented, reconstructed, and interpreted from incomplete, noisy, or indirect data. Addressing these questions has led to deep advances in areas such as functional and harmonic analysis, geometry, variational methods, inverse problems, partial differential equations, probability and statistics, and, more recently, learning theory. The resulting interplay between theory, computation, and applications has made imaging a uniquely fertile meeting point for pure and applied mathematics alike. Applications span an extraordinary range of domains, including biomedical and materials imaging, astronomy, environmental science, cultural heritage, and emerging technologies such as autonomous systems and data-driven diagnostics.
The rapid rise of artificial intelligence is now reshaping the landscape of imaging science once again. Data-driven approaches, in particular deep learning, have achieved remarkable empirical success in tasks such as reconstruction, segmentation, and synthesis. At the same time, their opaque nature and heavy reliance on data raise fundamental mathematical questions concerning stability, generalisation, interpretability, uncertainty quantification, and the incorporation of prior knowledge and physical constraints.
In this talk, I will present mathematical imaging as a continuing source of new mathematical ideas and challenges in the age of AI. I will highlight how modern approaches seek to blend data-driven models with structure, geometry, and physical principles, giving rise to novel analytical frameworks and computational paradigms. This perspective positions imaging not merely as an application area, but as a driver for the next generation of mathematical theory at the interface of analysis, computation, and learning.
This initiative is part of the “Ph.D. Lectures” activity of the project "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to Ph.D. students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.
Contatti:
laura.sangalli@polimi.it
paola.antonietti@polimi.it
Carola-Bibiane Schönlieb is Professor of Applied Mathematics at the University of Cambridge. There, she is head of the Cambridge Image Analysis group. Since 2011 she is a fellow of Jesus College Cambridge. Her current research interests focus on variational methods, partial differential equations and machine learning for image analysis, image processing and inverse imaging problems, and the mathematical foundations of machine learning. She has active interdisciplinary collaborations with clinicians, biologists and physicists on biomedical imaging topics, chemical engineers and plant scientists on image sensing, as well as collaborations with artists and art conservators on digital art restoration.
Her research has been acknowledged by scientific prizes, among them the LMS Whitehead Prize 2016, the Philip Leverhulme Prize in 2017, the Calderon Prize 2019, a Royal Society Wolfson fellowship in 2020, a doctorate honoris causa from the University of Klagenfurt in 2022, and by invitations to give invited lectures at several renowned applied mathematics conferences, including SIAM, ICM and ICIAM. She is also a SIAM fellow, and ELLIS fellow and a fellow of the Academy of the Mathematical Sciences.
Carola graduated from the Institute for Mathematics, University of Salzburg (Austria) in 2004. From 2004 to 2005 she held a teaching position in Salzburg. She received her PhD degree from the University of Cambridge (UK) in 2009. After one year of postdoctoral activity at the University of Göttingen (Germany), she became a Lecturer at Cambridge in 2010, promoted to Reader in 2015, promoted to Professor in 2018, and elected to the Professorship of Applied Mathematics (2006) in 2025. Carola convened the European Women in Mathematics Association between 2016 and 2020 and chaired the Committee for Applications and Interdisciplinary Relations (CAIR) of the EMS from 2021 to 2025. She also has been holding several leadership positions at SIAM.