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I am an assistant professor in Economics who did a masters in Mathematics in Cambridge (UK) some 10 years ago. I am an applied theorist, working at the intersection of psychology and economics - how to use insights from psychology to make ad hoc assumptions in economic models, where I use the literal meaning of ad hoc: "for this specific purpose". This leads to less grandiose predictions -- the assumptions are ad hoc and thus specific -- but hopefully also more useful.
My interest in category theory comes in two forms. First, I believe that it should be possible to leverage computers more in economics and other social sciences, in order to build more robust models (ones that are not so limited because we cannot solve them easily with pen and paper), and my feeling is that category theory might be of some use there. Second, I am interested in figuring out which questions are (in some sense that is nebulous to me currently) possible to answer meaningfully within certain classes of models or types of data, and which aren't -- questions of robustness of models. Some of that fits more naturally within statistics (sufficient statistics are a big thing in economics, especially the empirical parts), but I feel that category theory might help me think more clearly as well - or if not, it was worth a shot.