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I'm a neuroscience postdoc specializing in dynamical systems on networks and biophysical models of magnetic fields generated by brain activity. I think a useful, quantitative understanding of the brain will require a combination of abstract and concrete approaches, with useful stuff to be integrated all the way from semiring theory to the structure of ion channels.
I'm familiar with David Spivak's work, had the pleasure of bothering him as I was finishing up my dissertation, and think this latest book is far and away the most important introduction to the topic I've seen thus far. Starting from (pre)order is I think an optimal approach, and jumping almost immediately into Galois connections, which I think may be nice models for communication in general, brings things to some depth right away.
Beyond that, I have no idea what's going on. And although I have pressing time commitments at the moment (namely finishing a manuscript and hopefully finding a job), I'm going to try and follow along as best I can. I'll likely be late to the party for most if not all topics, so hopefully some of you revisit old posts once in awhile!