We consider the Landau-de Gennes variational model for nematic liquid crystal, in a 2-D stationary case. In particular, we aim to determine which phase (that is, uniaxial or biaxial) appears in the equilibrium configurations. We show that, in the low temperature range, minimizers are maximally biaxial, in the sense that they reach the maximum degree of biaxiality at some point. Next, we  discuss the asymptotic behavior of minimzers, as the elastic constant tends to zero, and prove the convergence to a locally harmonic map with singularities. Throughout the discussion, we will focus on the interplay between analysis and topology, arising in these problems.