This Wikipedia of Algebraic Geometry Will Forever Be Incomplete. That’s the Point.

Columbia professor Johan de Jong has spent the last 15 years gathering the foundational theorems of algebraic geometry in one place. His creation, the Stacks Project, offers a new model for organizing and visualizing mathematical information.

Kim Martineau
February 02, 2022

Early in the pandemic, the Stacks Project crossed a new milestone: 7,000 pages of proofs, theorems, lemmas, and sections. But unlike previous millennial markers, this one passed without celebration.

“Sad, but you know why,” Johan de Jong wrote in his blog. “The main new addition that got us over the hump was the chapter on derived categories of varieties. I hope you enjoy it.”

Stacks Project blog

And with that sunny update, the Columbia math professor was back at work, his eyes fixed on the next hump. Now 7,300 pages, the Stacks Project looks like a traditional textbook. But click into its 114 chapters, and equations and commentary spill down the screen, unfurling six decades of scholarship in bite-sized chunks.

The Stacks Project has become so popular with geometers—what mathematicians specializing in algebraic geometry call themselves—that it has inspired off-shoots in other fields. The site Kerodon is modeled after Stacks, and its creators hope it will do for topology what the Stacks Project has done for algebraic geometry. 

“The Stacks Project is a terrific new model for how to store and disseminate a large body of knowledge,” said Kerodon creator Jacob Lurie, a mathematician at the Institute for Advanced Study. “If you're looking for a reference or explanation for something that seems like it ought to be a standard fact, chances are that you can find it there.”

The American Mathematical Society recently awarded de Jong one of its highest honors, the Steele Prize, for his contributions to the field. 

“The material isn’t in a flat format as it has been for thousands of years,” said Kiran Kedlaya, a mathematics professor at University of California, San Diego. “Everything is broken into pieces, with links. This extra layer of organization allows you to search at random. It lowers the barriers, even for people working on this topic.”

“Without Johan there would be no Stacks Project.”

Max Lieblich, a mathematics professor at University of Washington and one of de Jong's former students

The casual reader will find few names with the familiar ring of a Pythagoras or Gauss. But the math laid out here is as embedded in daily life as triangles and normal curves; cryptography is based on the elliptic curves described by algebraic geometry; and street-mapping and face-recognition software use algebraic geometry to reconstruct the three-dimensional world on a flat screen. 

At its most basic, algebraic geometry is the study of circles and other simple shapes described by polynomial equations like x2 + y2 = 1. In the 1960s, French mathematician Alexandre Grothendieck laid the foundation for modern algebraic geometry with his text, Elements de Geometrie Algebriques. In it, he introduced his theory of schemes, a combination of algebra and geometry for describing shapes as ordinary as a circle or as unusual as a manifold with infinite dimensions defined by imaginary numbers. Algebraic stacks came along in the 1970s when mathematicians realized they needed an even more flexible way of classifying shapes in highly abstract situations. Stacks, for example, allowed them to calculate how many ways two triangles of the same shape might be arranged on a flat plane.

proof flowchart from the Stacks Project

But the theory of algebraic stacks was never spelled out as comprehensively as Grothendieck had done with schemes. This is the void that de Jong stepped in to fill in 2008. “I thought it would be good to collect all these results in one place so they’d be easier to reference,” he said.

Eleven years later, as de Jong typed in his last equation, he realized how much more was left to be done. Mistakes to fix. Definitions to align. Concepts to add and link. “To some extent, it’s just what I do now,” he said. “I sort of don’t know how to stop.” 

The enormity of the task is one factor fueling de Jong’s obsession. At 7,300 pages, the project still covers just half of the existing scholarship on algebraic stacks. The immediacy of online publishing is another motivating factor. Thanks to hyperlinks and the cloud, the Stacks Project is a living document that requires updates and corrections in the way plants require water and sunlight.  

“The Stacks Project is not static because algebraic geometry is not static,” said Kedlaya, a former student of de Jong’s. “The subject continues to expand, even at a foundational level. There’s constantly new material to incorporate, and this system is built to expand indefinitely.”

And the beauty of it is that no matter how big it grows, it’s still easy to search, thanks to a tagging system proposed by Cathy O’Neil, the best-selling author of Weapons of Math Destruction who happens to be married to de Jong. Tags not only let the Stacks Project expand indefinitely, they let mathematicians map concepts and see how they relate. 

A famous result in commutative algebra is mapped out in the Stacks Project: the logical implications of the proof of the Cohen structure theorem for complete local rings.

“You start to see the structure of mathematics emerging from this document,” said Max Lieblich, a mathematics professor at University of Washington.

Not everyone is equipped to take on a Sisyphean task of this magnitude. Colleagues describe de Jong as someone who is doggedly persistent and genuinely passionate about teaching, known for taking on up to four times as many graduate students as his peers and refusing to teach a course the same way twice; one year he taught a second-semester class on algebraic geometry as an Ask Me Anything series. That de Jong has tenure has also been integral to the project’s success, they say, freeing him from the academic grind of publishing.

Lieblich, another former student, is blunt: “Without Johan there would be no Stacks Project.”

And it’s de Jong’s exacting standards that also set the Stacks Project apart from other crowdsourced publications on the web. “Johan gets very mad when I call it Stackopedia,” said Kedlaya. “He reads every line that goes in.”

The one-editor model allows the Stacks Project to maintain one voice and a high level of quality control. But unlike the peer-reviewed literature that it attempts to corral into one place, the Stacks Project is designed to evolve. Long after de Jong is gone, this accumulation of knowledge will continue to grow.

“I want this thing to be alive and keep changing,” de Jong said.