I’m giving some talks at Oxford:

## 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.

If you’re nearby I hope you can come! All these talks will take place in Lecture Theatre B in the Computer Science Department—see the map below. Here are the times:

• Friday 21 February 2014, 2 pm: **Network Theory: overview**. See the slides or watch a video.

• Tuesday 25 February, 3:30 pm: **Network Theory I: electrical circuits and signal-flow graphs**. See the slides or watch a video.

• Tuesday 4 March, 3:30 pm: **Network Theory II: stochastic Petri nets, chemical reaction networks and Feynman diagrams**. See the slides or watch a video.

• Tuesday 11 March, 3:30 pm: **Network Theory III: Bayesian networks, information and entropy**. See the slides.

The first talk will be part of the OASIS series, meaning the “Oxford Advanced Seminar on Informatic Structures”.

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 surely somewhere in the Mathematical Institute building:

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

The Computer Science Department is shown in the map here:

The Mathematical Institute is a bit to the west:

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.

But there’s no need to wait: you can already read about various aspects of this stuff. Tobias Fritz and I have almost finished a paper giving a category-theoretic characterization of relative entropy, but you can read about it now:

• Relative entropy: Part 1, Part 2 and Part 3,

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:

• Categories in control.

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

• John Baez and Jacob Biamonte,

A Course on Quantum Techniques for Stochastic Mechanics.And people curious about the phylogenetic operad can read this:

• Operads and the tree of life,

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

Circuit diagrams now appear in the study of the Navier–Stokes equation, too.

Yeah, that’s cool! Unfortunately it’s for a modified (nonlocal) version of the Navier-Stokes equation that Terry Tao cooked up just to show that a large class of arguments to prove no-blowup theorems for the

actualequations can’t possibly work: the modified version obeys a bunch of the same inequalities, yet he can use it to simulate a circuit that blows up in finite time.You can see slides for two of my talks here:

• Network Theory: overview.

•Network Theory I: electrical circuits and signal-flow graphs.

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

Here are the slides for another talk:

• Network Theory II: stochastic Petri nets, chemical reaction networks and Feynman diagrams.

As always, complaints and questions are welcome!

A cute type: competitino –> competition. Or maybe you want to keep this one just for fun ;)

My student Blake Pollard caught that too, and suggested it could be a new typo of particle.

It must be one of the many decoy products of the typino then, of which we can see plenty in this tread!

The entry is actually off Parks Rd, not Keble Rd.

Thanks! Everyone used to go into the unofficial entrance off Keble Rd. However, it turns out that you need someone to let you in through a locked door no matter which way you enter. So, going in the official entrance and telling the person there that you want to attend the talk may be advisable. (Otherwise, just wait until someone walks through the door!)

Great talk! Really enjoyed it. I was surprised how much of it I followed, being a non-mathematician. (Must read up on category theory!) I shall make an effort to come to the other talks in the series.

I’m glad you liked it! The next ones will be bit a harder: if everything I say is easy to understand, they’ll kick me out of the academy.

Seriously, I need to show people that this isn’t “all sizzle and no steak”, and since I only have 3 hours to do it, I won’t be able to explain everything as gently as I’d like. But please come and see what you think.

Here is a video of a talk I gave yesterday, made by Brendan Fong. You can see the slides here—and then click the items in blue, and the pictures, for more information!

Here is a video of a talk I gave last Tuesday: “Network theory I: electrical circuits and signal-flow graphs”.

The talk here, part of a series, is an explanation of this viewpoint and how we can use it to take ideas from elementary particle physics and adapt them to study chemical reaction networks.

In the last of my Oxford talks I explain how entropy, relative entropy and Bayesian networks can be understood using certain categories related to probability theory.