While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. In this talk I will present an intrinsic characterization of those tensors that do, by means of polynomial equations of degree at most four. The exact degrees, and the corresponding polynomials, are different in each of three times two scenarios: ordinary, symmetric, or alternating tensors; and real-orthogonal versus complex-unitary. This is a joint project with J. Draisma, E. Horobet, and E. Robeva.

We start by discussing pushouts and (-1)- and 0-truncations, then we use these to discuss quotients of sets by mere relations and algebraic structures on sets. We finish with a particular case of the flattening lemma, which says that if W is a particular higher inductive type, then the total space of the fibration corresponding to a presheaf on W is equivalent to an appropriate higher inductive type.

The article is devoted to the proof of the existence of solutions of a system of integro-differential equations appearing in the case of anomalous diffusion when the negative Laplacian is raised to some fractional power. The argument relies on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains along with the Sobolev inequality for a fractional Laplace operator are being used.