You may be wondering, somewhere deep in the back of your mind, what happened to the Network Theory series on this blog. It’s nowhere near done! I plan to revive it, since soon I’ll be teaching a seminar on network theory at U.C. Riverside. It will start in October and go on at least until the end of March.

Here’s a version of the advertisement I sent to the math grad students:

### Network Theory Seminar

I’ll be running a seminar on Network Theory on Mondays from 3:10 to 4:30 pm in Surge 268 starting on October 6th.

Network theory uses the tools of modern math—categories, operads and more—to study complex systems made of interacting parts. The idea of this seminar is to start from scratch, explain what my grad students have been doing, and outline new projects that they and other students might want to work on. A lot has happened since I left town in January.

I hope to see you there!

If you want more detail, here is a sketch of what’s been happening.

1) Franciscus Rebro has been working on “cospans”, a very general way to treat a physical system with inputs and outputs and treat it as a morphism in category. This underlies all the other projects.

2) Brendan Fong, a student of mine at Oxford, is working on a category where the morphisms are electrical circuits, and composing morphisms is sticking together circuits:

• Brendan Fong, A compositional approach to control theory.

• John Baez and Brendan Fong, A compositional framework for passive linear networks.

The first paper here is an outline of the project; the second one actually carries out the project, or at least a large chunk of it.

3) Blake Pollard, a student in the physics department, has been studying Markov processes. In a Markov process, things randomly hop from one vertex of a graph to another along edges. Blake has created a category where morphisms are ‘open’ Markov process, in which things flow in and out of certain special vertices called ‘terminals’.

4) Jason Erbele has been working on categories in control theory, the branch of math used to study physical systems that interact with the outside world via inputs and outputs. After finishing this paper:

• John Baez and Brendan Fong, Categories in control.

he’s been taking more concept from control theory and formalizing them using categories.

5) Oh yeah, and what about me? I gave a series of lectures on network theory at Oxford, and you can see videos of them here:

• John Baez, Network Theory.

Jacob Biamonte and I have also more or less finished a book on chemical reaction networks and Petri nets:

• John Baez and Jacob Biamonte, *Quantum Techniques for Stochastic Mechanics*.

But I won’t be talking about that; I want to talk about the new work my students are doing!