Non-existence of orthogonal complex structures on the round 6-sphere

In this talk, we will review the famous Hopf problem, which dates back to 1948, of whether there is a complex structure on the round 6-sphere. Although some attempted answers have been advanced, both in the negative and in the positive directions, this problem is still open. We will review some of the work and partial results that appear in the literature and take a closer look at almost complex structures compatible with the metrics of constant sectional curvature.