Tensor- and spinor-valued random fields with applications to continuum physics and cosmology

In this paper, we review the history, current state-of-art, and physical applications of spectral theory two classes random functions. One class consists homogeneous isotropic fields defined on a Euclidean space taking values in real finite-dimensional linear space. to continuum physics, such field describes properties continuous medium situation, when microstructure is attached all points. The range fixed point set symmetry class, where compact Lie groups act by orthogonal representations. material group same at each acts trivially, while symmetries may nontrivially. an medium, rank 1 (resp., 2) correlation tensors transform under action according above representation its tensor square), making isotropic. Another cross-sections vector bundles over coset group. cosmology, models sky sphere, cross-section cosmic background. Cosmological Principle ensures that For convenience reader, necessary from multilinear algebra, theory, differential geometry reviewed Appendix.