## February 10, 2014

### Network Theory Talks at Oxford

#### Posted by John Baez

One of my dreams these days is to get people to apply modern math to ecology and biology, to help us design technologies that work with nature instead of against it. I call this dream ‘green mathematics’. But this will take some time to reach, since living systems are subtle, and most mathematicians are more familiar with physics.

So, I’ve been warming up by studying the mathematics of chemistry, evolutionary game theory, electrical engineering, control theory and information theory. There are a lot of ideas in common to all these fields, but making them clear requires some category theory. I call this project ‘network theory’. I’m giving some talks about it at Oxford.

(This diagram is written in Systems Biology Graphical Notation.)

Here’s the plan:

#### Network Theory

Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, signal-flow graphs, Bayesian networks, Feynman diagrams and the like. Mathematically minded people know that in principle these diagrams fit into a common framework: category theory. But we are still far from a unified theory of networks. After an overview, we will look at three portions of the jigsaw puzzle in three separate talks:

I. Electrical circuits and signal-flow graphs.

II. Stochastic Petri nets, chemical reaction networks and Feynman diagrams.

III. Bayesian networks, information and entropy.

All these talks will be in Lecture Theatre B of the Computer Science Department—you can see a map here, but the entrance is on Keble Road. Here are the times:

• Friday 21 February 2014, 2 pm: Network Theory: overview. Also available on YouTube.

• Tuesday 25 February, 3:30 pm: Network Theory I: electrical circuits and signal-flow graphs. Also available on YouTube.

• Tuesday 4 March, 3:30 pm: Network Theory II: stochastic Petri nets, chemical reaction networks and Feynman diagrams. Also available on YouTube.

• Tuesday 11 March, 3:30 pm: Network Theory III: Bayesian networks, information and entropy. Also available on YouTube.

I thank Samson Abramsky, Bob Coecke and Jamie Vicary of the Computer Science Department for inviting me, and Ulrike Tillmann and Minhyong Kim of the Mathematical Institute for helping me get set up. I also thank all the people who helped do the work I’ll be talking about, most notably Jacob Biamonte, Jason Erbele, Brendan Fong, Tobias Fritz, Tom Leinster, Tu Pham, and Franciscus Rebro.

Ulrike Tillmann has also kindly invited me to give a topology seminar:

#### Operads and the Tree of Life

Trees are not just combinatorial structures: they are also biological structures, both in the obvious way but also in the study of evolution. Starting from DNA samples from living species, biologists use increasingly sophisticated mathematical techniques to reconstruct the most likely “phylogenetic tree” describing how these species evolved from earlier ones. In their work on this subject, they have encountered an interesting example of an operad, which is obtained by applying a variant of the Boardmann–Vogt “W construction” to the operad for commutative monoids. The operations in this operad are labelled trees of a certain sort, and it plays a universal role in the study of stochastic processes that involve branching. It also shows up in tropical algebra. This talk is based on work in progress with Nina Otter.

I’m not sure exactly where this will take place, but probably somewhere in the Mathematical Institute, shown on this map. Here’s the time:

• Monday 24 February, 3:30 pm, Operads and the Tree of Life.

If you’re nearby, I hope you can come to some of these talks — and say hi!

(This diagram was drawn by Darwin.)

Posted at February 10, 2014 10:34 AM UTC

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### Re: Network Theory Talks at Oxford

All these great things have been happening at Oxford, and I keep missing them! First, the grand opening of the new site, and now this. And I’m extra sorry now that I’ve had a sneak preview of your and Tobias’s new paper :-)

Posted by: Tom Leinster on February 11, 2014 1:59 AM | Permalink | Reply to this

### Re: Network Theory Talks at Oxford

It looks like Brendan Fong has been volunteered to make videos of my talks — so with luck, people who want to see the talks can cut their carbon footprint and do it that way. I’ll definitely make my slides available. And I plan to blog about my paper with Tobias. In fact I already have, on Azimuth:

It actually makes sense to read these backwards, starting from Part 3, which states our main result.

People interested in control theory and categories can already see slides for the talk on that topic I gave here in Erlangen:

People interested in chemical reaction networks, Petri nets and Feynman diagrams can read this book:

So, there’s already a lot of gunk out there, but in my talks I plan to synthesize it, and I’ll make a webpage with the talks and links to all these references.

Posted by: John Baez on February 11, 2014 10:16 AM | Permalink | Reply to this

### Re: Network Theory Talks at Oxford

This post is a wonderful answer for those which tell us “you category jingos can’t do applied mathematics” :)

Just kidding: it’s awesome, I only feel guilty since I don’t have the means to attend… Can’t wait to see the videos.

Posted by: Fosco Loregian on February 11, 2014 5:43 PM | Permalink | Reply to this

### Re: Network Theory Talks at Oxford

Hi John,

I am looking forward to the slides and the videos.

The network literature is vast. One thing that strikes me that there seems to be a lot of different mathematical structures that are called “networks.” I will not try to get you to spell out a definition, but what is a network for you, roughly speaking?

Posted by: Eugene Lerman on February 11, 2014 6:55 PM | Permalink | Reply to this

### Re: Network Theory Talks at Oxford

Eugene wrote:

I will not try to get you to spell out a definition, but what is a network for you, roughly speaking?

Of course this is what the network theory project is supposed to figure out. It’s supposed to take the many things people call networks, create a mathematical taxonomy of them (or at least some of them), and prove general results about large classes of them, thereby ‘unifying’ the use of networks in different applied sciences.

But roughly I’d say this: for me, a network is a graph with labelled edges and/or vertices having some vertices designated as ‘inputs’ and some as ‘outputs’, viewed as a morphism in a bicategory. We typically get compact closed bicategories this way.

I guess my talks will explain that a bit better than I feel like doing now! But you can see the basic idea worked out in an example here:

• Brendan Fong, A a compositional approach to control theory.

Posted by: John Baez on February 12, 2014 8:27 AM | Permalink | Reply to this

### Re: Network Theory Talks at Oxford

You can see slides for two of my talks here:

I’ll keep cranking them out. As usual, I love questions and corrections!

Posted by: John Baez on February 17, 2014 2:21 PM | Permalink | Reply to this

### Re: Network Theory Talks at Oxford

Here are the slides for another talk:

As always, complaints and questions are welcome!

Posted by: John Baez on February 18, 2014 6:13 PM | Permalink | Reply to this

### Re: Network Theory Talks at Oxford

Here are the slides of my last talk, about work with Tobias Fritz and Tom Leinster… and some related work by Brendan Fong!

You can also see the first two talks on YouTube now (Network Theory Overview and Network Theory I.)

Posted by: John Baez on March 2, 2014 1:38 PM | Permalink | Reply to this

### Re: Network Theory Talks at Oxford

Now Network Theory II: stochastic Petri nets, chemical reaction networks and Feynman diagrams is on YouTube. I still have a long way to go to catch up with the Catsters, but I’m trying.

Posted by: John Baez on March 8, 2014 12:00 PM | Permalink | Reply to this

### Re: Network Theory Talks at Oxford

And now the last talk is on YouTube: Network Theory III. I thank Brendan Fong for videotaping all these and putting them up!

This last one was interrupted halfway through by a fire drill! — proof that the categorical approach to entropy is a hot topic.

Posted by: John Baez on March 16, 2014 10:18 PM | Permalink | Reply to this

### Re: Network Theory Talks at Oxford

By the way, in this talk I mistakenly said that relative entropy gave a continuous functor; in fact it’s just lower semicontinuous. I’ve fixed this now in the slides.

The last part of my talk was my own interpretation of Brendan Fong’s master’s thesis:

I took a slightly different approach, by saying that a causal theory $\mathcal{C}_G$ is the free category with products on certain objects and morphisms coming from a directed acyclic graph $G$. In his thesis he said $\mathcal{C}_G$ was the free symmetric monoidal category where each generating object is equipped with a cocommutative comonoid structure. This is close to a category with finite products, though perhaps not quite the same: a symmetric monoidal category where every object is equipped with a cocommutative comonoid structure in a natural way (i.e., making a bunch of squares commute) is a category with finite products. However, I believe this tiny difference makes no difference for what he does, because he is mainly concerned with symmetric monoidal functors from $\mathcal{C}_G$ into a category $\mathcal{D}$ with finite products, and here the cocommutative comonoid structure on any object of $\mathcal{C}_G$ is inevitably mapped to the unique cocommutative comonoid structure on the corresponding object of $\mathcal{D}$, which is automatically natural.

By making this slight change, I am claiming that causal theories can be seen as algebraic theories in the sense of Lawvere. This would be a very good thing, since we know a lot about those.

Posted by: John Baez on March 17, 2014 2:23 PM | Permalink | Reply to this

### Re: Network Theory Talks at Oxford

the last talk is on YouTube: Network Theory III.

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