Abstract: I will discuss universal Toda brackets due to Sagave. They are Mac Lane cohomology classes that determine Toda brackets in certain stable homotopy theories and provide an obstruction theory to the problem of realizing $\pi_* R$-modules as $R$-modules for a ring spectrum $R$.

Abstract: Since creation of quantum groups theory numerous attempts to elaborate an appropriate corresponding differential calculus were undertaken. Recently, a new type of noncommutative geometry has been obtained this way. Namely, we have succeeded in introducing the notions of partial derivatives on the enveloping algebras U(gl(m)) and constructing the corresponding de Rham complexes. All objects arising in our approach are deformations of their classical counterparts. In my talk I plan to introduce some basic notions of the Quantum Groups theory and to exhibit possible applications of this type Noncommutative Geometry to quantization of certain dynamical models.