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I apologize for asking a big list question, I've tried to avoid doing so for a while. I'll give my justification in a moment. The question is as follows: What are examples of strict applications ...

I am trying to wrap my head around all the different notions of dimension (and their equivalences). To get a sense of this, it would be nice to know the subtle difficulties that arise. I thus think it ...

If I were going to propose a new construction as a "replacement for resolution of singularities", what properties would my replacement have to have? [I am going to do no such thing -- this is purely ...

This is a major edit of the original post after receiving helpful comments. It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...