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Tomorrow my student Brandon Coya is defending his thesis at 11 am! Here is his thesis:

I'll blog about it later, but you can have a sneak preview now!

Abstract.We use the framework of "props" to study electrical circuits, signal-flow diagrams, and bond graphs. A prop is a strict symmetric monoidal category where the objects are natural numbers, with the tensor product of objects given by addition. In this approach, electrical circuits make up the morphisms in a prop, as do signal-flow diagrams and bond graphs. A network, such as an electrical circuit, with \(m\) inputs and \(n\) outputs is a morphism from \(m\) to \(n\), while putting networks together in series is composition, and setting them side by side is tensoring. Here we work out the details of this approach for various kinds of electrical circuits, then signal-flow diagrams, and then bond graphs. Each kind of network corresponds to a mathematically natural prop. We also describe the "behavior" of electrical circuits, bond graphs, and signal-flow diagrams using morphisms between props. To assign a behavior to a network we "black box" the network, which forgets its inner workings and records only the relation it imposes between inputs and outputs. The process of black-boxing a network then corresponds to a morphism between props. Interestingly, there are two different behaviors for any bond graph, and we show that the relationship between these two behaviors arises as a natural transformation. To achieve all of this we first prove some foundational results about props. These results let us describe any prop in terms of generators and equations, and also define morphisms of props by naming where the generators go and checking that relevant equations hold. Technically, the key tools are the Rosebrugh–Sabadini–Walters result relating circuits to special commutative Frobenius monoids, the monadic adjunction between props and signatures, and a result saying which symmetric monoidal categories are equivalent to props.

## Comments

There's a famous paper by G Oster, Perelson and Katchalsky in Quarterly Rev Biophysics 1973 on 'network thermodynamics' which uses bond graphs-like 130 pages--i couldn't get through it. I like this applied category course but i have trouble with all the terminology--eg 'prop'---i'm just used to normal basic erdos-renyi-bollobos graph theory (sometimes applied as 'topochemistry') . some of this has been discussed on azimuth blog. i wonder if all these lectures and discussions will still be available online on azimuth. (i'm having trouble keeping up ).

`There's a famous paper by G Oster, Perelson and Katchalsky in Quarterly Rev Biophysics 1973 on 'network thermodynamics' which uses bond graphs-like 130 pages--i couldn't get through it. I like this applied category course but i have trouble with all the terminology--eg 'prop'---i'm just used to normal basic erdos-renyi-bollobos graph theory (sometimes applied as 'topochemistry') . some of this has been discussed on azimuth blog. i wonder if all these lectures and discussions will still be available online on azimuth. (i'm having trouble keeping up ).`

Good luck to Coya and if possible to share here the presentation slides and or video , it would be great. Yes Mart.. terminology/notation is a important barrier for me too..my strategy is bet on patience and keep myself in touch with the theme. Maybe a also a directed multigraph approach could help ..but its not clear to me if and the exact specific situations that it could be corrected used and if it really holds indeed..

`Good luck to Coya and if possible to share here the presentation slides and or video , it would be great. Yes Mart.. terminology/notation is a important barrier for me too..my strategy is bet on patience and keep myself in touch with the theme. Maybe a also a directed multigraph approach could help ..but its not clear to me if and the exact specific situations that it could be corrected used and if it really holds indeed..`

Mart: props are defined in Brandon's thesis - in fact they're defined in the abstract you just read. But if you don't know about symmetric monoidal categories you'll have to learn about those first; they're fundamental in most of applied category theory. You can learn about them here:

Physics, topology, logic and computation: a Rosetta Stone.and you will also learn about them in this course. Looking at the pictures in Brandon's thesis, and this paper, will give you an idea of why these ideas are important.

`Mart: props are defined in Brandon's thesis - in fact they're defined in the abstract you just read. But if you don't know about symmetric monoidal categories you'll have to learn about those first; they're fundamental in most of applied category theory. You can learn about them here: * John Baez and Mike Stay, _[Physics, topology, logic and computation: a Rosetta Stone](http://arxiv.org/abs/0903.0340)_. and you will also learn about them in this course. Looking at the pictures in Brandon's thesis, and this paper, will give you an idea of why these ideas are important.`

Pierre - I'll post slides of Brandon's thesis defense, but of course not videos: giving a thesis defense is nerve-racking enough without being filmed!

`Pierre - I'll post slides of Brandon's thesis defense, but of course not videos: giving a thesis defense is nerve-racking enough without being filmed! <img src = "http://math.ucr.edu/home/baez/emoticons/tongue2.gif">`

:) true!!

`:) true!!`

The slides are here:

`The slides are here: * Brandon Coya, <a href = "thesis_defense_coya.pdf">Dissertation defense</a>, May 15, 2018.`

John, that link seems to be broken :( It takes me back to this very forum thread.

`John, that link seems to be broken :( It takes me back to this very forum thread.`

Yes Jonathan.. broken to me too

`Yes Jonathan.. broken to me too`