Symplectic nilmanifolds and applications

The talk is devoted to the remarkable towers of bundles
$$M^n \to M^{n-1} \to \cdots \to S^1, N \ge 2$$
with fiber the circle $S^1$. This towers are dened by the nilpotent groups of the polynomial transformations of the real line.
Each $M^n, n \ge 2$, is a smooth nilmanifold with a 2-form which gives a symplectic structure on any $M^{2k}$.

Such manifolds play an important role in dierent areas of mathematics. We will discuss the differential-geometric and algebro-topologic results and unsolved problems, concerning this manifolds.