21 November 2017:
This week's - or really last week's - progress:
1) Daniel got this paper accepted by TAC
, subject to very minor revisions. It's a great paper and you should all read it - I tend to assume you all read each other's papers to keep up with what's happening, but I fear that's not true. By now I realize this paper is just the beginning of using topos theory to study networks:
Abstract. We introduce the notion of a span of cospans and define, for them, horizonal and vertical composition. When in a topos C, these compositions satisfy the interchange law. A bicategory is then constructed from C-objects, C-cospans, and doubly monic spans of C-cospans. The primary motivation for this construction is an application to graph rewriting.
By the way, Daniel, you should fix the typo in the first sentence of the abstract! Also I think "these compositions satisfy the interchange law" is false unless the spans have monic legs. When you make all the fixes please put a new version on the abstract, and please send me a copy.
2) I gave the weekly General Biology Seminar at Caltech, and gave this talk:
Abstract. If biology is the study of self-replicating entities, and we want to understand the role of information, it makes sense to see how information theory is connected to the 'replicator equation' — a simple model of population dynamics for self-replicating entities. The relevant concept of information turns out to be the information of one probability distribution relative to another, also known as the Kullback–Leibler divergence. Using this we can get a new outlook on free energy, see evolution as a learning process, and give a clearer, more general formulation of Fisher's fundamental theorem of natural selection.
It was a bit scary talking to a large room of biologists about biology, but it went well. I spent the day talking to lots of people, mainly associated to Eric Winfree's gang — he studies the computational power of chemical reaction networks. For example, he found a chemical reaction network with two species such that in equilibrium the probability that it has any number of the two species, plotted in gray-scale, looks like a cartoon of Darth Vader. (He can get any probability distribution he wants.)
There are a lot of people at Caltech working on artificial neural networks and also on control theory in biology. I should get to know some of them!