Mathematician Susanna Zimmermann, winner of the Latsis Swiss Science Prize 2026, is advancing the field of birational geometry, exploring the basic transformations that explain how objects can change while preserving the same deeper structure.
Many people find mathematics difficult. For Susanna Zimmermann, it is more like a game: There is a well-defined problem, no predetermined strategy, and the challenge of finding a completely new solution to reach a clear and conclusive answer. “Then it is time for the next problem,” says the professor of mathematics at the University of Basel, who has been awarded the Latsis Prize 2026 for her groundbreaking research in algebraic geometry.
“For me, research is a very creative process. I enjoy thinking about a problem and finding a solution in a structured and logical way. I have to break the problem down into its essential elements, put them together in a different way, and somehow arrive at a solution. The more abstract mathematics becomes, the more creative this process can be,” she shares.
Since humanity introduced idealised shapes such as squares, triangles and spheres, constant efforts have been made to understand their properties, how they can be transformed, and what the consequences of those transformations are. This process dates back to the ancient Greek mathematician Euclid and has become increasingly sophisticated over time, extending to highly abstract geometries that are hard to visualise. Much of mathematics progresses by introducing a variation to an existing concept and exploring which properties remain valid and which new ones emerge.
A central problem in geometry is deciding when two shapes should be considered the same, since this allows mathematicians to classify them and identify the properties that truly define them. Is a sphere, for example, more similar to a cube or to a doughnut? The answer depends on which transformations are allowed in the system under consideration.
A triangle does not become a different triangle simply because it is moved, rotated or reflected in a mirror. But in ordinary geometry, it does change if one of its angles is altered or if it is elongated in only one direction. In other contexts, stretching is allowed, so triangles with different proportions can be treated as equivalent.
In topology, the branch of geometry that studies shapes under stretching and bending without cutting or joining them, even a triangle and a circle are considered equivalent because both form a single closed boundary with no holes. For the same reason, a sphere and a cube are equivalent, while a doughnut is different because it has a hole. Modern geometry applies these ideas to far more complex spaces, including spaces with more than three dimensions.
Susanna Zimmermann specialises in birational geometry, a branch of mathematics that studies transformations between geometrical objects that can be reversed. “The idea is that you have some geometric objects and you want to understand which can be transformed into one another and which cannot,” she says.
For example, almost all the points on a sphere can be matched with points on a flat plane, and the correspondence can then be reversed to recover the sphere. The two objects look very different, but they still share the same underlying algebraic structure. This makes it possible to study a complicated shape through a simpler one without losing the information needed to reconstruct it.
Besides advancing our understanding of pure mathematics, this type of research provides the foundation for future applications. One such example is to assess complex artificial intelligence models such as neural networks, where different internal settings can sometimes produce the same result. These many-to-one relationships can make it difficult to evaluate a model’s performance properly. Birational methods help researchers compare AI models and estimate how well they will perform on new data.
Another application is in cryptography, where researchers are exploring encryption keys based on geometric transformations. The original information can then be recovered by applying the inverse transformation used during encryption.
To decide whether two shapes are truly different, mathematicians study the transformations that preserve them, called symmetries, and the properties that remain unchanged under those transformations. “The symmetries between surfaces can become very complicated, but they are all built from very simple symmetries, such as rotations,” explains Susanna Zimmermann. Her research focuses on identifying the fundamental building blocks of these more complex symmetries.
Among all birational symmetries, she is particularly interested in those that rearrange a given geometric space without altering its core structure. For example, straight lines can be mapped to curved lines while preserving enough information to reconstruct the original configuration. All such transformations form what mathematicians call the Cremona group, a set of birational symmetries named after the nineteenth-century Italian mathematician Luigi Cremona. Studying a Cremona group means understanding all the different ways a given space can change while remaining, at a deeper algebraic level, the same.
Since Cremona introduced this idea in 1863, these symmetries have been studied continuously, and the last decade has brought significant advances. Yet their fundamental building blocks in more than three dimensions are still unknown. “This lack of understanding is a bit embarrassing for the field. I would love to find these building blocks, and that’s where my research is going.”
In 2021, Susanna Zimmermann achieved a breakthrough that had a vast impact on the field. One of the central questions in the study of Cremona groups is whether they contain smaller subgroups with a meaningful internal structure, which had only been proved in two dimensions. She decomposed complicated birational transformations into simpler geometric steps and tracked how often particular types of steps occur. This technique allowed her to prove that Cremona groups also contain such subgroups in three dimensions and higher.
This seminal paper “really opened a lot of doors,” she says, and was the result of an intense collaboration with other researchers. While some mathematicians often work alone and publish single-author papers, she is in constant contact with people from around the world, and these collaborations frequently lead to new results. “I may have a problem to solve and be halfway to the answer. Then I meet another person who has been thinking about the same question but has the other half of the solution. We join forces and finish it together,” she says.
Mathematics is regarded as the most objective of the sciences, yet before reaching a rigorous proof, researchers can have very different perspectives on the same problem.
“The way we think is very personal, and it is fascinating to discover someone else’s point of view. It can add an entirely new layer of understanding,” she concludes. For Susanna Zimmermann, mathematical problems are games, and the best ones are not played alone.
Susanna Zimmermann grew up in the canton of Glarus and studied mathematics at the University of Basel. First drawn to algebra, she soon discovered a passion for geometry, a combination that led her to algebraic geometry. After completing her master’s thesis in 2013, she chose to pursue a PhD in Basel, which she defended in 2016, on questions that still shape her research today.
An SNSF-funded postdoctoral stay in Toulouse ensued, followed by successive professor positions in Angers, Paris-Saclay and, since February 2025, back at the University of Basel. She received the CNRS Bronze Medal in 2020 and the ICBS Frontiers of Science Award in 2024.
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