We give a complete description of gradings by abelian groups on certain classes of nilpotent Lie algebras, including Lie algebras of maximal class. This has consequences concerning symmetries on homogeneous spaces of nilpotent Lie groups. These results are joint with Elizabeth Remm and Michel Goze (Univesite de Haut Alsace, France).

In this talk on a more classical part of ergodic theory, that of Z-actions, I will explain the construction of rank-one transformations via cutting and stacking that goes back to von Neumann and Kakutani and has been used to create examples and counterexamples of various mixing-like properties. Following the explanation of the subject, I will present some of my work on when such transformations are mixing. Some of the results presented are joint work with Cesar Silva.