The variety of complete quadrics, which is used by Schubert in his famous computation of the number of space quadrics tangent to 9 quadrics in general position, is a particular compactification of the space of non-singular quadric hypersurfaces in n dimensional complex projective space.

In this talk, towards a theory of Springer fibers for complete quadrics, I will describe our recent work on the unipotent invariant complete quadrics. These results involve interesting combinatorics, and in particular, give a new q-analog of Fibonacci numbers.

This is joint work with Michael Joyce.