  We survey some of the AGT relations between N=2 gauge theories in four
dimensions and geometric representations of symmetry algebras of
two-dimensional conformal field theory on the equivariant cohomology of their
instanton moduli spaces. We treat the cases of gauge theories on both flat
space and ALE spaces in some detail, and with emphasis on the implications
arising from embedding them into supersymmetric theories in six dimensions.
Along the way we construct new toric noncommutative ALE spaces using the
general theory of complex algebraic deformations of toric varieties, and
indicate how to generalise the construction of instanton moduli spaces. We also
compute the equivariant partition functions of topologically twisted
six-dimensional Yang-Mills theory with maximal supersymmetry in a general
Omega-background, and use the construction to obtain novel reductions to
theories in four dimensions.
