In the past decade, theory of Kuranishi structures on moduli space of pseudo-holomorphic curves has been in the center of debates between some mathematicians in the field of symplectic geometry.

Kenji Fukaya is teaching a course on this subject at Simons Center and the course is being recorded. Here is the link to the first lecture. http://scgp.stonybrook.edu/archives/10004

In order to clarify some comments and claims, for myself and may be others, I will gradually post some questions on mathoverflow. Hopefully, I will gather the results of these discussions in a Lecture note.

1) At some point (min 33-34), John Morgan comments that for an arbitrary compact subset of Euclidean space, $K\subset \mathbb{R}^m$, and for small enough open set $U\subset \mathbb{R}^m$, $U\cap K$ can be realized as the zero set of some smooth map $f\colon U \to \mathbb{R}^n$. How is the proof? For example if this compact set if the contour set in $\mathbb{R}$.

Comment: zero locus of smooth maps + extra structure, are the bulding blocks of Kuranishi spaces

2) A Delign-Mumford stack is somehow a category itself. Then, is there a category of Deligne-Mumford stacks that includes fiber products, ...?

Comment: A Dream of Kuranishi theory is to build a category out of Kuranihsi spaces.