That's almost exactly the kind of thing I would have liked to do---something Very applied to all kinds of things (as is done to some extent by people at SFI , as well math sociologists, psychologists, economists, econophysicists etc.)
I probably was not PhD level material (though i passed a couple grad school classes as an undergrad) but if you are lucky (unlike me--too stupid to even use what luck i have--win the lottery and either have your jackpot stolen or squander it since that is what your peers tell you to do even if you'd prefer to do something else ) there is a place for people not doing research and college teaching. (You can teach at lower levels which i did--but i had same issues as in academia --personal conflicts, narrow focus,
biases and prejudice---so i quit. In age of internet you can go independent and get everything online (except personal and financial support. I'm no Ramanujan but he's an example of what can happen with no support. Julian Barbour managed to find a side gig and be independent. ).
Most of what i could get when involved with academia was stuff like modeling conformations of DNA , and stuff related to biotech. Thats fine and itslef can be interesting and sometimes useful (but in this age of 'Big Pharma' and the opiate epidemic , and obesity fueled by junk food, you can question exactly how valuable biomedical research focusing on genes and biomolecules is---studying prejudice, blame , responsibility and priviledge may be more socially valuable, along with environment, behavioral economics (alot of which now is applied by corporations for marketing) , inequality, intolerance, etc.
In my academic world these topics were basically taboo----people are too busy saving the world to deal with 'victimology'---those are the whiners and complainers we are trying to help through our research. They say 'We Gotta save the (or at least my world)---publish or perish'. 'All boats will rise , and when they do we will start to feed, medicate , educate and find jobs for the whining starving babies, who will then be 100 years old if still alive. 'all happy together, so goes the weather, forever'.
I wonder if the application of category theory to social issues or problems leads to anything new and/or applicable. Maybe all you get is an elegant, aesthetically attractive reformulation of things like 'theory of war, class struggle and conflict ' (eg Louis Fry Richardson, Goodwin --both pioneers of nonlinear dynamics) written in math language as opposed to words.
I see this is from Chicago. My area (Wash DC) has a few gun shootings and homicides every week, Baltimore has maybe 1 /day, and south side chicago has more than that. I bet one could categorize this issue. The set of people who go to U Chicago is not the same as the set who live a few blocks from there.