higher geometry / derived geometry
higher topos theory
geometric little (∞,1)-toposes
geometry (for structured (∞,1)-toposes)
geometric big (∞,1)-toposes
function algebras on ∞-stacks
loop space object, free loop space object
fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos / of a locally ∞-connected (∞,1)-topos
derived algebraic geometry
étale (∞,1)-site, Hochschild cohomology of dg-algebras
schematic homotopy type
derived noncommutative geometry
derived smooth geometry
differential geometry, differential topology
derived smooth manifold, dg-manifold
smooth ∞-groupoid, ∞-Lie algebroid
higher symplectic geometry
higher Klein geometry
higher Cartan geometry
Jones' theorem, Deligne-Kontsevich conjecture
Tannaka duality for geometric stacks
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An Artin stack is a geometric stack over the étale site: the stack quotient of a scheme whose automorphism groups are algebraic groups.
This is the general notion of algebraic stack. But often the latter term is used in a more restrictive sense to mean Deligne-Mumford stack.
Artin stacks are named after Michael Artin.
See for instance
Gérard Laumon, L. Moret-Bailly, Champs algébriques , A series of Modern Surveys in Mathematics vol. 39, Springer-Verlag 2000.
Bertrand Toen, Master course on algebraic stacks.
Dennis Gaitsgory, Nick Rozenblyum, A study in derived algebraic geometry, Section I.2.4.