Turned out the reason he was teaching a course that semester was that he was really excited to share a new idea he'd had about how to visualize the complex projective plane. There were two (or three?) students in attendance.
Okay, here's Calabi's visualization of the complex projective plane. The complex projective plane (as everyone knows) is the set of equivalence classes of complex 3-vectors, where two vectors are equivalent if one is a complex scalar multiple of the other. Calabi's visualization: for a given representative, take the real part and imaginary part of that 3-vector separately, now that's a pair of vectors in real 3-space, and those are much easier to visualize. Choose a particular representative for the equivalence class: use up the dilation part of your complex scalar to normalize your 3-vectors, then use up the rotation part to make them orthogonal. Now notice that multiplication by a unit-length complex scalar will rotate the real and imaginary 3-vectors through the plane they span, so we can visualize an element of the complex projective plane as an oriented ellipse in real 3-space.
Actually we have some singular ones — if the real and imaginary parts are the same length (and orthogonal) then instead of ellipses we have circles, and if the real or imaginary part is zero then we have line segments. But we have a 1-parameter family of oriented ellipses (the parameter is the eccentricity) where at each parameter we have a quotient of SO(3) (it's almost the special orthogonal group but not quite) and at one end of the family it reduces to a 2-sphere and at the other end to a real projective plane.
But the main point is instead of having to visualize 3 or 2 complex dimensions (6 or 4 real dimensions) now we get to visualize ellipses (1 real dimension!) inside real 3-space. Much more accessible to geometric intuition.