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it is really exciting to be on this Forum and I hope to share a lot of ideas, work or even just suggestions with you.
I am a first year PhD student in Mathematics at Rome (Tor Vergata), working in Operator Algebras and Category Theory and their applications in Mathematical Physics, my main interest is the concept of Entropy and I am working on it in two projects: the first one is in AQFT on curved spacetimes and the other is in categorification of noncommutative probability and the notion of entropy therein.
I arrived here thanks to the work of John Baez: in it I saw beautiful applications of the abstract study of symmetries in other fields, not just plain abstract/esthetical applications but rather also actual applications which shed light upon the 'mathematisation' of concepts in different areas.
My personal approach with trying to think a little bit outside the box, which is something that I would like to deepen here, began with the study of the Kleinian approach to geometry (https://ncatlab.org/nlab/show/Erlangen+program) which I find essentially multi-disciplinary and a possible mathematical-context creator tool when analyzing or modeling mathematically unknown phenomena.
Some times ago, I found a similar approach in the book by N. N. Čencov "Statistical Decision Rules and Optimal Inference" and the field of Information Geometry provides other intuitions. Also in QFT there are "informationally-based" approach to derive the spacetime geometry from information theoretic quantities.
This might be possible also for complex phenomena, with an underlying internal geometry emerging from their information content: this is really what I want to focus on in this forum.
So let me know what do you think and if you have hints, references, etc...