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Hi! I am a physics undergraduate studying in Mumbai. I was introduced to applied category theory thorough John's numerous posts on Google Plus especially the one on coarse-graining and the one on props. (I just started reading the book a few days ago and am beginning to realize the journey is going to be much more difficult!)
Anyway I would be looking to gain insights, from as much as I am able to learn, for possible applications in quantum information/ computation for which I noticed there is a thread already but not much activity. Also all the applications that you guys come up with are surely going to be interesting.
Then towards a more vague direction, I am very intrigued by the coarse-graining idea: extracting relevant and coarse information from an actually more complicated and detailed model.
I believe that is what machine learning algorithms are actually doing even though most of the model seem ad hoc (to me). I wonder if category theory/information theory can provide rigor and direction to that area.
It seems it can even tackle what is called "emergent behavior" - for example how classical behavior emerges out from quantum physics: what features are lost, which are retained? How macroeconomics emerges from the more rigorous microeconomics?
Then, about the theory of knowledge itself. As John says we need many models of different scope and accuracy and we need to get systematic about dealing with all these models and how they are related. It is a common feature of science that models have implicit assumptions built in them that are not supported by measurement (for example Newton's gravity) that need to be overthrown and perhaps we can come up with a more systematic representation of models respecting the order of magnitude to which observations have been tested.
Also, there's the simple structure of concepts: the hierarchy of say: coffee-table > table > furniture or man > animal > living-thing > object and how these different concepts relate. How our sentences carry/lose meanings or create "generative effects".
If you think about it almost every scientific problem or artistic endeavor has a structure to it and that is how I got interested in this course. Thanks to John for running it! Thank you for reading.