I’m visiting Erlangen from now until the end of May, since my wife got a grant to do research here. I’m trying to get a lot of papers finished. But today I’m giving a talk in the math department of the university here, which with Germanic brevity is called the Friedrich-Alexander-Universität Erlangen-Nürnberg.
You can see my slides here, or maybe even come to my talk:
The title is a pun. It’s about categories in control theory, the branch of engineering that studies dynamical systems with inputs and outputs, and how to optimize their behavior.
Control theorists often describe these systems using signal-flow graphs. Here is a very rough schematic signal-flow graph, describing the all-important concept of a ‘feedback loop’:
Here is a detailed one, describing a specific device called a servo:
The device is shown on top, and the signal-flow graph describing its behavior is at bottom. For details, click on the picture.
Now, if you have a drop of category-theorist’s blood in your veins, you’ll look at this signal-flow graph and think my god, that’s a string diagram for a morphism in a monoidal category!
And you’d be right. But if you want to learn what that means, and why it matters, read my talk slides!
The slides should make sense if you’re a mathematician, but maybe not otherwise. So, here’s the executive summary. The same sort of super-abstract math that handles things like Feynman diagrams:
also handles signal-flow graphs. The details are different in important and fascinating ways, and this is what I’m mainly concerned with. But we now understand how signal-flow graphs fit into the general theory of networks. This means we can proceed to use modern math to study them—and their relation to other kinds of networks, like electrical circuit diagrams:
Thanks to the Azimuth Project team, my graduate students and many other folks, the dream of network theory as a step toward ‘green mathematics’ seems to be coming true! There’s a vast amount left to be done, so I’d have trouble convincing a skeptic, but I feel the project has turned a corner. I now feel in my bones that it’s going to work: we’ll eventually develop a language for biology and ecology based in part on category theory.
So, I think it’s a good time to explain all the various aspects of this project that have been cooking away—some quite visibly, but others on secret back burners:
• Jacob Biamonte and I have written a book on Petri nets and chemical reaction networks. You may have seen parts of this on the blog. We started this project at the Centre for Quantum Technologies, but now he’s working at the Institute for Scientific Interchange, in Turin—and collaborating with people there on various aspects of network theory.
• Brendan Fong is working with me on electrical circuits. You may know him for his posts here on chemical reaction networks. He’s now a grad student in computer science at Oxford.
• Jason Erbele, a math grad student at U.C. Riverside, is working with me on control theory. This work is the main topic of my talk—but I also sketch how it ties together with what Brendan is doing. There’s a lot more to say here.
• Tobias Fritz, a postdoc at the Perimeter Institute, is working with me on category-theoretic aspects of information theory. We published a paper on entropy with Tom Leinster, and we’ve got a followup on relative entropy that’s almost done. I should be working on it right this instant! But for now, read the series of posts here on Azimuth: Relative Entropy Part 1, Part 2 and Part 3.
• Brendan Fong has also done some great work on Bayesian networks, using ideas that connect nicely to what Tobias and I are doing.
• Tu Pham and Franciscus Rebro are working on the math that underlies all these projects: bicategories of spans.
The computer science department at Oxford is a great place for category theory and diagrammatic reasoning, thanks to the presence of Samson Abramsky, Bob Coecke and others. I’m going to visit them from February 21 to March 14. It seems like a good time to give a series of talks on this stuff. So, stay tuned! I’ll try to make slides available here.