  Two-dimensional (2-D) incompressible, inviscid fluids produce fascinating
patterns of swirling motion. How and why the patterns emerge are long-standing
questions, first addressed in the 19th century by Helmholtz, Kirchhoff, and
Kelvin. Countless researchers have since contributed to innovative techniques
and results, but the overarching problem of swirling 2-D motion and its
long-time behavior remains largely open. Here we advocate an alternative
view-point that sheds light on this problem via a link to isospectral matrix
flows. The link is established through V. Zeitlin's beautiful model for the
numerical discretization of Euler's equations in 2-D. When considered on the
sphere, Zeitlin's model enables a deep connection between 2-D hydrodynamics and
unitary representation theory of Lie algebras as pursued in quantum theory.
Consequently, it provides a dictionary that maps hydrodynamical concepts to
matrix Lie theory, which in turn gives connections to matrix factorizations,
random matrices, and integrability theory, for example. The transferal of
outcomes, from finite-dimensional matrices to infinite-dimensional fluids, is
then supported by hard-fought results in quantization theory -- the field which
describes the limit between quantum and classical physics. We demonstrate how
the dictionary is constructed and how it unveils techniques for 2-D
hydrodynamics. We also give accompanying convergence results for Zeitlin's
model on the sphere.
