  We summarise some recent results on solution landscapes for two-dimensional
(2D) problems in the Landau--de Gennes theory for nematic liquid crystals. We
study energy-minimizing and non energy-minimizing solutions of the
Euler--Lagrange equations associated with a reduced Landau-de Gennes free
energy on 2D domains with Dirichlet tangent boundary conditions. We review
results on the multiplicity and regularity of solutions in distinguished
asymptotic limits, using variational methods, methods from the theory of
nonlinear partial differential equations, combinatorial arguments and
scientific computation. The results beautifully canvass the competing effects
of geometry (shape, size and symmetry), material anisotropy, and the symmetry
of the model itself, illustrating the tremendous possibilities for exotic
ordering transitions in 2D frameworks.
