Mathematician Katharina Hübner explores objects from a mysterious numerical world that reveal themselves only through fascinating and unexpected detours.
There are places no one will ever visit, simply because they don’t exist in our everyday reality. Mathematicians in the research project CRC/TRR 326 GAUS explore such places daily: they investigate fascinating objects that can neither be touched nor drawn and that no one can truly imagine. To make this possible, they must get creative and approach these inaccessible objects through something they can grasp: they construct models of the objects, study the resulting structures, and, through this indirect method, uncover the properties of the original forms – properties they could never have identified directly.
An Artful Touch
Let’s take a simple example: imagine a circle, and let’s assume that this thin line, which loops around once and returns to its origin, is too complex to study directly. The clever approach Katharina Hübner employs is to construct layers above this shape: she takes a string that extends infinitely in both directions and winds it up like a spool. You can picture the resulting helix as a parking garage ramp that spirals upward evenly, round after round, with no end. This spiral is positioned precisely above the original circle so that its shadow, when illuminated from above, perfectly traces the circle’s line. “We have now created a more accessible object,” Katharina Hübner explains, “one that still describes the circle with precision.” After all, when the spiral is unwound and straightened, it becomes a simple straight line – an object as straightforward as can be. Yet, in the way it is wound, it also carries all the information about the circle on which it is based.
Such constructions are called coverings in mathematics. They may sound technical, but they are based on an intuitive principle: if an object is too complex to handle directly, you look at a simpler object above it and use its structure to understand the one below. Even though a spiral looks different from a circle, it reveals that the circle has a hole in the middle – and it does so in a manner that can be calculated and generalized. “You can understand many structures better by looking at the space above them,” says Katharina Hübner.
The appearance of a covering is no coincidence – it is a kind of fingerprint of the space beneath it. Mathematicians have built an entire theory around this idea. They analyze all possible coverings of a space, uncovering insights about its structure: How many holes does it have? How is it organized? How do paths run through it? This theory is powerful and works beautifully as long as we stay within the familiar world of real numbers.
Katharina Hübner isn’t studying circles, though. “For such a straightforward object, the theoretical effort would simply be overkill,” she says. She’s interested in objects that are far more complex. To understand what she’s working with, it helps to take a brief detour through equations. The simple circle we just considered can be described by the polynomial equation x² + y² = 1, since all points (x, y) that satisfy this equation together form a circle with a radius of 1 around the origin (0, 0). If you choose different coefficients and look at an equation like 4x² + 7y² = 3, you get an ellipse. But what happens if the coordinates (x, y) that solve such an equation don’t come from the familiar world of real numbers, but from a completely different number system?
Curious Numbers
An intriguing alternative is the so-called p-adic numbers, for which p represents any prime number, such as 3, 5, or 41. Completely different neighborhood relationships apply in this world. As such, in the p-adic world defined by the prime number p = 5, numbers like 33, 36, and 76 are considered the same size, while 75 is smaller than 30. It may sound entirely absurd, but it follows clear rules and is therefore a legitimate way to measure sizes and distances – even if it contradicts our usual intuition.
Using such unusual numbers has far-reaching consequences for the geometry of spaces. For example, if you take the familiar circle equation x² + y² = 1 and look for solutions in the p-adic numbers, you obtain what is known as a rigid analytic space. If you consider a circular ring – a disk with a hole in the middle – in p-adic terms, a different rigid analytic space emerges. On paper, these spaces resemble their classical counterparts, such as a circle or a ring, but their internal logic is unfamiliar. This unfamiliarity makes them difficult to understand and, at the same time, deeply fascinating.
For centuries, number theory was considered the quintessential example of useless mathematics (…) until it became clear that numbers theory structures are indispensable for protecting our digital communication from hackers. What may seem abstract today could suddenly become essential tomorrow.
The central question Katharina Hübner is exploring is whether the theory of coverings also works in this other world. In other words, can the coverings of a rigid analytic space be used just as elegantly to determine how many holes it has and what holds it together at its core? The answer is more complicated than expected. “When you apply the classical theory to rigid analytic spaces, you do get coverings, but far too many,” the mathematician explains. “They are also highly branched and difficult to control, and they do not provide the information we are looking for.” However, Katharina Hübner has developed methods to keep these coverings in check and to use this layered structure to uncover insights about the rigid analytic spaces beneath them.
Connections Between Geometry and Numbers Theory
The principle of comparing a complex object that cannot be directly grasped with a simpler one that shares its local structure but is easier to understand globally is called uniformization. The unwrapped line is the uniformization of the circle. While the line is simple – it has no holes and is not closed – it still carries the complexity of the circle in the symmetry of the unwrapped spiral. And in mathematics, symmetries can be studied using powerful tools. The principle of uniformization underpins the research within the CRC/TRR 326 GAUS project, “Geometry and Arithmetic of Uniformized Structures.” Katharina Hübner and her co-authors Jakob Stix, Piotr Achinger, and Marcin Lara, together with many colleagues from the universities of Frankfurt, Darmstadt, Heidelberg, Mainz, Münster, and Hanover, are investigating how uniformization works in different mathematical worlds and exploring the deep connections between geometry and numbers theory hidden within these structures. To do so, they draw on cutting-edge techniques, including groundbreaking results by Fields Medalist Peter Scholze from Bonn, who has built an unexpected bridge between classical geometry and a parallel mathematical world in which new symmetries become visible.
To explore abstract objects more closely, mathematicians often take unusual detours and employ remarkable techniques. Since practical applications for uniformization do not yet exist, these methods do not make bridges more stable or improve smartphone batteries. Instead, the work in the CRC/TRR 326 GAUS project is fundamental research: scientists develop understanding long before it is clear how it might be applied. Number theory, for example, was long considered the quintessential example of “useless” mathematics – a playground for abstract thought, as beautiful as a poem but just as practical – until it became clear that its structures are indispensable for securing digital communication. What seems abstract today may become essential tomorrow. Katharina Hübner and her colleagues work on such questions at the very edge of what can be conceived, where familiar tools fail and new ones must be invented.
Author: Aeneas Rooch
The Collaborative Research Centre / Transregio 326 GAUS – a project by the Rhine-Main Universities and their partners
Since 2021, the CRC/TRR 326 “Geometry and Arithmetic of Uniformized Structures (GAUS)” has been exploring how highly complex geometric and arithmetic structures can be described using simpler spaces. Goethe University Frankfurt serves as the coordinating university; other university members are Technical University of Darmstadt and Heidelberg University. The project’s partners include Johannes Gutenberg University Mainz – which, together with Frankfurt and Darmstadt, forms the Rhine-Main Universities (RMU) network – as well as Leibniz University Hannover and the University of Münster.
