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The original development of general equilibrium theories involved differential topology.

I wonder if there are any recently developed theories, in any field of economic theories, that utilize differential topology (DT).

I've seen some recent papers but none of them used DT. Some real analysis, topological topology, and convex analysis seem enough.

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The main reason differential topology had some success in economics is that supplies powerful methods to show that something holds generically, mainly Sard's theorem and the transversality theorem. Some of these methods have been generalized to contexts without differentiability, see for example the paper "A Prevalent Transversality Theorem for Lipschitz Functions" by Shannon. Differential topology is still used in general equilibrium theory, you might take a look at recent work of Yves Balasko.

In positive political theory, there are results to the effect that generically no majority winner exists in multidimensional spatial voting models. A good introduction to this topic is the book "Positive Political Theory I" by Austen-Smith and Banks, the definitive treatment is the paper "The generic existence of a core for q-rules" by Saari.

Inspired by the work on regular equilibria in general equilibrium theory, regular Nash equilibria were also studied in game theory, starting with Harsanyi. An elegant approach can be found in the paper "The theory of normal form games from the differentiable viewpoint" by Ritzberger (preprint here). Many genericity results in general equilibrium have corresponding versions in game theory, for example, "most" normal form games have a finite and odd number of Nash equilibria.

Another area where differential topology is used is in the study of stability with respect to some dynamics. Indeed, one can define regular equilibria via the replicator dynamic from evolutionary game theory (that is how Ritzberger does it). For a recent example of using some differential topology (index theory, treat also in more generality), see the working paper "The Index +1 Principle" by McLennan.

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  • $\begingroup$ Thanks a lot. You must be a econ prof. I have a book "equilibrium manifolds" by Balasko. Would you like to give us some professors working with the same approach as Balasko? $\endgroup$ – High GPA Aug 1 '17 at 5:27
  • $\begingroup$ I am not a professor, but I am an academic economist. A good way to find people working on a topic is by putting relevant publications into Google Scholar and looking who is citing it. For example, Andrea Loi and Stefano Matta have written papers on the equilibrium manifold. I don't think it is a very active topic of research though. $\endgroup$ – Michael Greinecker Aug 1 '17 at 8:07
  • $\begingroup$ Thanks again. I believe it is not hot because of caviar to the general. Is GE theory still the crown jewelry? $\endgroup$ – High GPA Aug 1 '17 at 8:25
  • $\begingroup$ Not many people work on general equilibrium nowadays. The most active area is probably connected to finance. $\endgroup$ – Michael Greinecker Aug 1 '17 at 8:32
  • $\begingroup$ Oh, my master major is in finance and econ. If you are also in finance related field, do you need hard-working assistants or collaborators now or in future by any chance? Looking at your "measure theory" tags, you must be working on contracts with asymmetric info, right? $\endgroup$ – High GPA Aug 1 '17 at 8:44

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