I review two classes of strong coupling problems in condensed matter physics,
and describe insights gained by application of the AdS/CFT correspondence. The
first class concerns non-zero temperature dynamics and transport in the
vicinity of quantum critical points described by relativistic field theories. I
describe how relativistic structures arise in models of physical interest,
present results for their quantum critical crossover functions and
magneto-thermoelectric hydrodynamics. The second class concerns symmetry
breaking transitions of two-dimensional systems in the presence of gapless
electronic excitations at isolated points or along lines (i.e. Fermi surfaces)
in the Brillouin zone. I describe the scaling structure of a recent theory of
the Ising-nematic transition in metals, and discuss its possible connection to
theories of Fermi surfaces obtained from simple AdS duals.Comment: 39 pages, 12 figures; Lectures at the 5th Aegean summer school, "From
gravity to thermal gauge theories: the AdS/CFT correspondence", and the De
Sitter Lecture Series in Theoretical Physics 2009, University of Groninge