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Starting with the complexes, quaternions and the octonions we study five families of groups and homogeneous spaces each building geometrically on the previous families. For each family a particular theorem is studied from the perspective of the structure developed. The evolution of the spaces and the corresponding results form a sort of mathematical memoir of a career spanning more than 50 years. [ABSTRACT FROM AUTHOR]

We study (homogeneous and inhomogeneous) anisotropic Besov spaces associated to expansive dilation matrices A ∈ GL (d , R) , with the goal of clarifying when two such matrices induce the same scale of Besov spaces. For this purpose, we first establish that anisotropic Besov spaces have an alternative description as decomposition spaces. This result allows to relate properties of function spaces to combinatorial properties of the underlying coverings. This principle is applied to the question of classifying dilation matrices. It turns out that the scales of homogeneous and inhomogeneous Besov spaces differ in the way they depend on the dilation matrix: Two matrices A , B that induce the same scale of homogeneous Besov spaces also induce the same scale of inhomogeneous spaces, but the converse of this statement is generally false. We give a complete characterization of the different types of equivalence in terms of the Jordan normal forms of A , B. [ABSTRACT FROM AUTHOR]

We study nontrivial entropy invariants in the class of parabolic flows on homogeneous spaces, quasi-unipotent flows. We show that topological complexity (i.e., slow entropy) can be computed directly from the Jordan block structure of the adjoint representation. Moreover using uniform polynomial shearing we are able to show that the metric orbit growth (i.e., slow entropy) coincides with the topological one for quasi-unipotent flows (this also applies to the non-compact case). Our results also apply to sequence entropy. We establish criterion for a system to have trivial topological complexity and give some examples in which the measure-theoretic and topological complexities do not coincide for uniquely ergodic systems, violating the intuition of the classical variational principle. [ABSTRACT FROM AUTHOR]