Topological concepts in economic theory QUESTION: What are the major or systematic applications of post-1960s mathematics to microeconomics? 


For example, in the late 19th century, Fisher first used the mathematical ideas of Gibbs to construct modern utility theory. In the 20th century, Mas-Colell incorporated topological ideas to study general equilibrium. What about the late 20th, early 21st century? 

For example, consider directed graph theory, measure theory, topology, the category theory and modern homology or cohomology, topos methods, functional integration, etc.  

Note 1: econometrics/statistics, without modelling, is excluded. The only modern mathematics used there is random walk theory, and the ergodic problem, solved via complex analysis. RW and EP are not specific to economics.

Any appropriate economics publication is an answer. This included also those published in non-strictly economics journals, e.g. the Journal of Mathematical Psychology.

Note 2:
Yes, I know, this type of work is rarer (not to be confused with obscurity: some of it is well known). That is what makes it easy to miss such a reference when it is published. Hence the question.
I strongly suspect that an emerging important area for applications of measure theory will be in approximate dynamic programming techniques. Approximate dynamic programming (aka "reinforcement learning" in the computer science literature) has been the direction of research work in the last ~10-20 years of the dynamic programming literature. Economics is only just now starting to adopt some of these advances. For example of the direction of the DP literature, see Bertsekas' most recent 4th edition expansion of his dynamic programming series, or Powell's Approximate DP: Solving the Curse of Dimensionality. Economists are just starting to pick up some of these tools, both directly and indirectly, and I suspect that they will have a growing impact on the literature over the next few years. Some of the analytical background for convergence of these methods is topology and dynamical systems.

A good example of theoretical contribution to this type of literature from economists is Pál and Stachurski (2013), Fitted Value Function Iteration With Probability One Contractions (ungated version here). Peruse that paper and you can see the importance of a good grasp of measure theory. Stachurski's book Economic Dynamics is actually a very nice exposition of dynamic programming from this perspective, building at a pace which works for multiple levels of graduate student/professional (measure theory comes in formally at the end I believe -- I'm still working towards those insights). 

Hopefully this answers your question to some degree. I'm afraid that the phrase "post-1960s mathematics" is somewhat ambiguous to me (due to my own lack of knowledge of history of maths literature), so if I've completely missed the mark, my apologies!