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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Given a topological group or algebraic group or Lie group, etc., GG, a homogeneous GG-space is a topological space or scheme, or smooth manifold etc. with transitive GG-action.
A principal homogeneous GG-space is the total space of a GG-torsor over a point.
There are generalizations, e.g. the quantum homogeneous space for the case of quantum groups.
A special case of homogeneous spaces are coset spaces arising from the quotient G/HG/H of a group GG by a subgroup. For the case of Lie groups this is also called Klein geometry.
Specifically for GG a compact Lie group and T↪GT\hookrightarrow G a maximal torus, then the coset G/TG/T play a central role in representation theory and cohomology, for instance in the splitting principle.
In analysis and number theory, certain functions on certain coset spaces play a role as automorphic forms (e.g. modular forms). See there for more.
Under weak topological conditions (cf. Hegason), every topological homogeneous space MM is isomorphic to a coset space G/HG/H for a closed subgroup H⊂GH\subset G (the stabilizer of a fixed point in XX).
Grassmannian, flag variety