Finite step Nilspaces are compact spaces $X$ together with closed
collections of "cubes" '$C^{n}(X)\subset X^{2^{n}}$, $n=1,2,\ldots$
verifying some natural axioms. These objects arise "Higher Order
Fourier Analysis" in Additive Combinatorics as well as in Topological
Dynamics and Ergodic Theory under the guise of maximal nilfactors. A
fundamental theorem of Camarena and B. Szegedy states that a finite
step nilspace is a nilmanifold, i.e. a quotient of a nilpotent Lie
group with a discrete cocompact group. We will give a new proof of
this theorem. The talk will not assume any familiarity with the
subject.