I’ve been rather quiet about Azimuth projects lately, because I’ve been too busy actually working on them. Here’s some of what’s happening:
• Jason Erbele is finishing his thesis, entitled Categories in Control: Applied PROPs. He successfully gave his thesis defense on Wednesday June 8th, but he needs to polish it up some more. Building on the material in our paper “Categories in control”, he’s defined a category where the morphisms are signal flow diagrams. But interestingly, not all the diagrams you can draw are actually considered useful in control theory! So he’s also found a subcategory where the morphisms are the ‘good’ signal flow diagrams, the ones control theorists like. For these he studies familiar concepts like controllability and observability. When his thesis is done I’ll announce it here.
• Brendan Fong is also finishing his thesis, called The Algebra of Open and Interconnected Systems. Brendan has already created a powerful formalism for studying open systems: the decorated cospan formalism. We’ve applied it to two examples: electrical circuits and Markov processes. Lately he’s been developing the formalism further, and this will appear in his thesis. Again, I’ll talk about it when he’s done!
• Blake Pollard and I are writing a paper called “A compositional framework for open chemical reaction networks”. Here we take our work on Markov processes and throw in two new ingredients: dynamics and nonlinearity. Of course Markov processes have a dynamics, but in our previous paper when we ‘black-boxed’ them to study their external behaviour, we got a relation between flows and populations in equilibrium. Now we explain how to handle nonequilibrium situations as well.
• Brandon Coya, Franciscus Rebro and I are writing a paper that might be called “The algebra of networks”. I’m not completely sure of the title, nor who the authors will be: Brendan Fong may also be a coauthor. But the paper explores the technology of PROPs as a tool for describing networks. As an application, we’ll give a new shorter proof of the functoriality of black-boxing for electrical circuits. This new proof also applies to nonlinear circuits. I’m really excited about how the theory of PROPs, first introduced in algebraic topology, is catching fire with all the new applications to network theory.
I expect all these projects to be done by the end of the summer. Near the end of June I’ll go to the Centre for Quantum Technologies, in Singapore. This will be my last summer there. My main job will be to finish up the two papers that I’m supposed to be writing.
There’s another paper that’s already done:
• Kenny Courser has written a paper “A bicategory of decorated cospans“, pushing Brendan’s framework from categories to bicategories. I’ll explain this very soon here on this blog! One goal is to understand things like the coarse-graining of open systems: that is, the process of replacing a detailed description by a less detailed description. Since we treat open systems as morphisms, coarse-graining is something that goes from one morphism to another, so it’s naturally treated as a 2-morphism in a bicategory.
So, I’ve got a lot of new ideas to explain here, and I’ll start soon! I also want to get deeper into systems biology.
In the fall I’ve got a couple of short trips lined up:
• Monday November 14 – Friday November 18, 2016 – I’ve been invited by Yoav Kallus to visit the Santa Fe Institute. From the 16th to 18th I’ll attend a workshop on Statistical Physics, Information Processing and Biology.
• Monday December 5 – Friday December 9 – I’ve been invited to Berkeley for a workshop on Compositionality at the Simons Institute for the Theory of Computing, organized by Samson Abramsky, Lucien Hardy, and Michael Mislove. ‘Compositionality’ is a name for how you describe the behavior of a big complicated system in terms of the behaviors of its parts, so this is closely connected to my dream of studying open systems by treating them as morphisms that can be composed to form bigger open systems.
Here’s the announcement:
The compositional description of complex objects is a fundamental feature of the logical structure of computation. The use of logical languages in database theory and in algorithmic and finite model theory provides a basic level of compositionality, but establishing systematic relationships between compositional descriptions and complexity remains elusive. Compositional models of probabilistic systems and languages have been developed, but inferring probabilistic properties of systems in a compositional fashion is an important challenge. In quantum computation, the phenomenon of entanglement poses a challenge at a fundamental level to the scope of compositional descriptions. At the same time, compositionally has been proposed as a fundamental principle for the development of physical theories. This workshop will focus on the common structures and methods centered on compositionality that run through all these areas.
I’ll say more about both these workshops when they take place.