Visualizing Alexander Grothendieck

Kind of amazing:

Starting in the 1950s, Alexander Grothendieck revolutionized math by introducing many new concepts: schemes, stacks, motives, topoi and more. He wrote over 6000 pages! And then, after many quarrels with the mathematical establishment of France, he disappeared into the Pyrenees, where he now lives in seclusion.

The Stacks Project is an open-source reference book with many authors which aims to explain a lot of the math Grothendieck and his collaborators created. It’s currently 4000 pages long! You can download it as a single huge PDF file… but much better, you can read it on the web, and see how each result relies on previous ones!

4 days ago, +Pieter Belmans made this a lot more fun. You can now see beautiful pictures of this enormous network of mathematics! For example, the picture below is everything leading up to “result 01WC” which says that “a locally projective morphism is proper”.

I don’t even know what this means… but never mind. What’s cool is that you can go to a page with this picture:

http://stacks.math.columbia.edu/tag/01WC/graph/force

and however your mouse over any dot and see the different results leading up to this one!

Someday, if civilization doesn’t collapse, all math will be linked up this way. There will be tools to help you explore it, starting with the easy stuff – and links to videos where people explain the ideas! There may even be online cafes where you can hang out and talk to other travelers who are visiting the same branch of math:

“Well, I came here because I was learning about Coxeter groups and someone told me I had to learn about root systems of Lie algebras. I plan to spend a few days here and then go back – do you want come along?”

But even right now, people are already using these networks to study statistical properties of the structure of mathematics! For example, +Cathy O’Neil, aka “mathbabe”, has gone through all 10,445 items in the Stacks Project. For each item, she looked at the graph like the one here. The nodes are the all the items on which the given item depends. The edges are the links between those items. Then she plotted the number of edges as a function of the number of nodes. You can see it here:

Analyzing the complexity of the Stacks Project graphs

What does it mean? Cathy has some ideas…

… but I think we’re starting to see a new kind of metamathematics, where people use statistical methods to study the structure of mathematics itself. This is mathematics as actually done by people, so it involves issues of taste and style. These are subjective things. But I suspect there are some features of math that are fairly independent of who is doing it. Maybe some theorems are ‘important’ in a fairly objective sense – important crossroads that most travelers tend to stop at. And someday we may understand why.

For more on the Stacks Project’s new network visualization tools, read this blog article by Cathy O’Neil:

http://mathbabe.org/2013/07/30/the-stacks-project-gets-ever-awesomer-with-new-viz/