Since its founding, K-theory has presented seemingly intractable obstacles to computation. 

K-theory is a branch of pure mathematics that studies rings, algebraic structures that index systems of numbers using two binary operations. K-theory assigns rings a series of invariants, called K-groups, that provide insight into their character. 

Invariants are vital to understanding core truths and relationships in mathematics. They describe properties that remain constant through alteration, shedding light on – or sparking curiosity about – a wide range of objects in algebra, geometry, and topology.

Yet while K-theory has flourished and grown since American mathematician Daniel Quillen’s Fields-medal-winning definition in the 1970s, many simple rings have resisted computation. Notable among these are rings of the form Z/p^n (z mod p to the n), which have been a focus of K-theory since its beginning. It has been proven that the K-groups of these rings can, in principle, be computed. But it would entail levels of complexity and abstraction so elevated that such calculations have been considered impossible.

Until now. Mathematicians Benjamin Antieau and Achim Krause have designed an algorithm capable of computing K-groups. With co-author Thomas Nikolaus, they debut two new studies: one that outlines a streamlined algebraic approach to this mathematical gridlock and another that presents the algorithm made possible by this approach, along with a surprising conjecture illuminated by the newly possible computation.

“K-theory is a process of transformation that takes as input something we understand very well and gives as output something that we would like to understand. These rings that come up in modular arithmetic – where you add, multiply, and then reduce modulo some integer – are relatively simple. With K-groups, they transform into fascinating answers that are wildly complicated. But since K-groups are defined in a way that makes them impossible to compute in practice, no one has been able to explore that process. Now, we can,” said Antieau. 

 
The Key to the Algorithm? A Purely Algebraic Object

Antieau and Krause refined and completed their algorithm at the Institute of Advanced Study (IAS) in 2024, where they participated in the School of Mathematics’ special year on p-adic arithmetic geometry. Like K-theory, this area of inquiry uses finely grained methods to study algebraic and geometric structures. It can intersect with and supplement K-theory, especially when applied to rings.

IAS’ choice to focus the 2023-2024 special year on p-adic arithmetic geometry was motivated by a decade of remarkable advances in the field. These advances have led to breakthroughs in multiple areas of mathematics, including the 2019 introduction of prismatic cohomology by Bhargav Bhatt, Matthew Morrow, and Peter Scholze. This, through a variant called syntomic cohomology, provides an algebraic take on K-theory.

“Quillen defined K-theory in a way that requires you to go through topology. Bhatt and Scholze discovered a purely algebraic object that offers new methods to study K-theory without the topological complications that make computation unrealistic. In the end, prismatic cohomology gave us enough leverage to boil down the computation down to relatively small matrices – which, in this case, is still tens of thousands of entries,” said Krause.

The reduction in computational cost is massive. Antieau and Krause’s work restructured something that no amount of time, money, or hardware could ever possibly compute into something that could be handled by a laptop computer.

“In the case we focus on, the invariants that K-theory gives you are all finite groups. Because they are all finite, you can ask questions about them and get answers. What group are they? How many elements do they have? For each of the rings we consider, there is a sequence of K-groups to investigate,” said Krause.

While prismatic cohomology created the tools these researchers needed, it did not afford a clear path to applying those tools. Between 2019 and 2024, Antieau and Krause partnered to adapt them for algorithm design. 

“One of the biggest things we had to overcome in that gestational period was that the prismatic cohomology was not finite in the answer for the rings that concerned us. It could not be stored in a computer in a meaningful way,” said Antieau.

Prismatic cohomology is algebraic. Unlike topology, it can be computed with elements, which gives it an advantage over Quillen’s formulation. 

However, in this case, it provided an infinite algebraic object, which presented a new obstacle to overcome. Needing to abridge this object, Antieau and Krause adapted syntomic cohomology to compute only the portion of this infinite algebraic object that contributed to the final answer. 

“What we achieved was really on the borderline of what was theoretically possible,” said Antieau.

In residence at IAS, Antieau and Krause put the finishing touches on their algorithm. The answers it delivered were rich with information. 

The groups the program computed all began with a familiar region of disorder. But as the computation continued, chaos transformed into regularity. Patterns emerged. 

Before, no computation had gotten far enough to where this change occurred. Suddenly, there was empirical evidence of something new.

“We realized that if you go far enough out, half the groups become trivial. They vanish. We called this our ‘even-vanishing conjecture’ because the ones in the even position all happened to be trivial,” said Krause. 

Within weeks of formulating the conjecture, the authors discovered a proof of the theorem, using the same machinery that powers the algorithm itself.

Because the K-groups vanished, Antieau and Krause were able to compute the exact number of elements in all the nearby groups. For the first time, they were able to provide comprehensive answers to questions about these rings.

“These insights were made possible by the experimental math. By getting to see the number of examples this algorithm was able to generate – much more than anyone has previously been able to see – we have been able to make educated guesses and then prove that these guesses are theorems,” said Antieau.

“It’s been a real joy to create a new technological tool to allow for experimentation in a field – this border between algebra and topology – that doesn’t have any traditional experimental tools,” said Krause.

The value of their intervention is clear. 

“Bhatt and Scholze made the problem algebraic. But we made it finite,” said Krause.