The lecture will be about complete embedded minimal surfaces in R^3 (some immersed ones help the explanations). Before 1978
very little was known and after 1984 progress became rapid. I will
illustrate with pictures, how observed features of known surfaces
led to new, increasingly abstract, constructions. I will not assume
that the audience consists of minimal surface experts, the lecture
is intended to be understandable by graduate students.

In this talk I present my recent results on the regularity conditions for a solution to the 3D Navier-Stokes equations with powers of the Laplacian, which incorporates the vorticity direction and its magnitude simultaneously. For the proof of the we exploit geometric properties of the vortex stretching term as well as the estimate using the Triebel-Lizorkin type of norms.