  We revisit the duality between five-dimensional supersymmetric gauge theories
and deformations of two-dimensional Yang-Mills theory from a new perspective.
We give a unified treatment of supersymmetric gauge theories in three and five
dimensions using cohomological localization techniques and the Atiyah-Singer
index theorem. We survey various known results in a unified framework and
provide simplified derivations of localization formulas, as well as various
extensions including the case of irregular Seifert fibrations. We describe the
reductions to four-dimensional gauge theories, and give an extensive
description of the dual two-dimensional Yang-Mills theory when the
three-dimensional part of the geometry is a squashed three-sphere, including
its extension to non-zero area, and a detailed analysis of the resulting matrix
model. The squashing parameter $b$ yields a further deformation of the usual
$q$-deformation of two-dimensional Yang-Mills theory, which for rational values
$b^2=p/s$ yields a new correspondence with Chern-Simons theory on lens spaces
$L(p,s)$.
