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Two Researchers Are Rebuilding Mathematics From the Ground Up
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foundations of mathematics
Two Researchers Are Rebuilding Mathematics From the Ground Up
By
Konstantin Kakaes
May 20, 2026
By replacing the most fundamental concept in topology, Peter Scholze and Dustin Clausen are taking the first step in a far bigger program to understand why numbers behave the way they do.
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Kristina Armitage/Quanta Magazine
By Konstantin Kakaes
Contributing Writer
May 20, 2026
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algebra
category theory
foundations of mathematics
fractals
history of science
mathematics
set theory
topology
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Let’s start with what’s probably the most tired, overused joke in math: A topologist is someone who can’t tell a coffee cup from a doughnut. Both, you see, have a hole in them.
Topology is usually described as a sort of “rubber sheet” geometry in which two shapes are considered the same if one can be stretched or compressed into the other without tearing it. But this summary leaves out something essential: How do topologists, and the many other mathematicians who rely on their methods, rigorously account for all this stretching? They don’t look at a doughnut and a coffee cup, squint, and say to themselves, “Sure, I can intuitively see how to squeeze one into the other, so they must be the same.” Rather, they describe a shape in a way that can “forget” about distance while respecting the underlying structure in a flexible way, allowing it to bend and stretch.
When these “topological spaces” were developed over 100 years ago, they played a major part in the revolutions in logic and set theory that marked the boundary between 19th-century and modern mathematics. Their birth was a crucial waypoint on math’s inexorable march from the numbers and shapes that people encounter in everyday life into ever more abstract caverns of thought. Topological spaces have since become the foundation for huge chunks of mathematics. If you think of math as a skyscraper, topological spaces are concrete pilings, driven deep into the bedrock of common sense that all of math ultimately rests on.
But disconcertingly, topological spaces turn out to be extremely poorly suited for a big chunk of modern math: They are an awkward setting in which to do algebra, which is something mathematicians quite like doing.
For years, mathematicians figured they just had to live with the limitations of topological spaces. If you’re working on the 87th story of a skyscraper, fixing the foundations in the subbasement is a scary proposition.
Peter Scholze prefers coming up with new definitions rather than coming up with new proofs. As he put it, he’s “trying to give names to what is there.”
Barbara Frommann/Hausdorff Center for Mathematics, University of Bonn
But over the past decade, Peter Scholze of the Max Planck Institute for Mathematics in Bonn and Dustin Clausen of the Institute of Advanced Scientific Studies in France have sought to replace topological spaces. They have defined a new category of mathematical objects called condensed sets, which resemble a sort of infinitely fine dust and retain all the nicest properties of topological spaces without the drawbacks. Dust, it turns out, is a better foundational material than the pebbly, well-understood soil of topological spaces.
“They are solving a problem we didn’t know we had,” said Ravi Vakil, a mathematician at Stanford University and president of the American Mathematical Society, “because we already had what we thought were reasonable solutions.” As a result, “a whole slate of mathematics has become much simpler.”
It’s an ambitious project. The new definitions and concepts that Scholze and Clausen have introduced are powerful but also complicated and hard to learn. Scholze, for his part, is not sure how widely used they will become. On the other hand, he sees them as just the first step in a far bigger program to understand why numbers behave the way they do.
Doing math can be a little like rock climbing: Just as the route you take up a sheer face can incorporate creativity and even elegance in the way it sequences technical maneuvers, so too can a proof. Both traverse existing terrain. Most research — even some of the best research — takes the form of finding new routes to known peaks. But in mathematics, there is a weird relationship between the equipment and the...