higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
derived smooth geometry
Given a space $X$ of sorts, in particular a scheme or variety, with singularities, i.e. with a subspace $S \subset X$ at which the geometry is non-regular (specifically: not smooth), a resolution of the singularity is a suitable regular (non-singular) space $\widehat X$ equipped with a morphism back to the original space
which is an isomorphism away from the singular locus.
Typical resolution of singularities is by “blow-up” of the singularity where the singular point is replaced by an n-sphere/projective space (and its neighbourhood by a tautological line bundle), then called the “exceptional divisor” of the blow-up.
(quick review of the basic details includes Berghoff 14, section 4.1)
the Fulton-MacPherson compactification of configuration spaces of points may be regarded as resolution of the fat diagonal inside an iterated Cartesian product;
this generalizes to “wonderful compactifications” resolving singularities given by more general subspace arrangements
Bridgeland stability over resolution of ADE-singularities: see there
The existence of resolutions of singularities by “blow-up” was established, for ground fields of characteristic zero, in some generality in
Basic review:
Marko Berghoff, section 4 of: Wonderful renormalization, 2014 (pdf, doi:10.18452/17160)
Loring Tu, Section 29.2 of: Introductory Lectures on Equivariant Cohomology, Annals of Mathematics Studies 204, AMS 2020 (ISBN:9780691191744)
The theorem of Hironaka 64 was used to discuss singular distributions (in the sense of generalized functions) in
This method is closely related to the resolution of singularities of propagators/Feynman amplitudes by passage to compactified configuration spaces of points, as disucussed at Feynman amplitudes on compactified configuration spaces of points.
See also
Wikipedia, Resolution of singularities
Wikipedia, Blowing up
Wikipedia, Exceptional divisor