# Commutative and non-commutative algebra

Tensors are widely used mathematical objects, defined in the context of multilinear algebra. The problem of writing a tensor as the sum of elementary tensors can be studied in different ways and, in the relevant case of symmetric tensors, commutative algebra and the theory of apolarity play a crucial role. One of the research activities of the group in commutative alegbra deals with this problem, that is with the sum of powers decomposition of homogeneous polynomials.

Our group is also interested in spaces of matrices of constant rank, which can be interpreted as matrices of linear forms whose kernel and cokernel are vector spaces varying smoothly on the projective space, that is, vector bundles. The connection with commutative algebra arises from the study of the graded modules of sections of these bundles, and of their minimal free resolutions.