We present a homogeneous real analytic hypersurface in C^{3}, two-nondegenerate, uniformly Levi degenerate of rank one, with a seven-dimensional CR automorphism group such that the isotropy group of each point is two-dimensional and commutative. The classical tube Γ_{C}over the two-dimensional real cone in R^{3}is also homogeneous and has a seven-dimensional CR automorphism group. However, our example is$not$biholomorphic to the tube over the real cone, because the two-dimensional isotropy groups of Γ_{C}are, in contrast, noncommutative.

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A similar formula to the one established by Ansemil and Floret for symmetric tensor products of direct sums is proved for alternating and Jacobian tensor products. It is then applied to stable spaces where a number of isomorphisms between spaces of tensors or multilinear forms are unveiled. A connection between these problems and irreducible group representations is made.

Quantum Fourier analysis is a subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We establish bounds on the quantum Fourier transform F, as a map between suitably defined L^p spaces, leading to an uncertainty principle for relative entropy. We cite several applications of quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a topological inequality, and we outline several open problems.

We consider subordination chains of simply connected domains with smooth boundaries in the complex plane. Such chains admit Hamiltonian and Lagrangian interpretations through the Löwner–Kufarev evolution equations. The action functional is constructed and its time variation is obtained. It represents the infinitesimal version of the action of the Virasoro–Bott group over the space of analytic univalent functions.