Anagrafe della ricerca

Tensors are higher dimensional matrices and nowadays are ubiquitous throughout science. As for matrices, they are universally useful tools to collect and represent data of any kind. During the last decades, a remarkable amount of activity has been devoted to establishing foundational results on tensors, which in turn are motivated by applications in several areas such as algebraic statistics and complexity of matrix multiplication. Nevertheless, several basic questions remain elusive to this date. The situation is even more convoluted when tensors are defined over the field of real numbers. As a result, research on tensors is blossoming in several directions importing techniques from disparate fields such as algebraic geometry, representation theory, and combinatorics. Decompositions of tensors are convenient and compact ways to express them. Our project aims to study tensor decompositions from a geometric perspective through algebraic geometry and commutative algebra, and consists of four main research directions: P1. Dimensions of secant varieties and general ranks: the knowledge of dimensions of secant varieties allows us to understand when they fill the whole space of tensors of a given format. This says what is the rank of a generic tensor of that format. P2. Spaces of decompositions: It is interesting to study properties shared by all minimal decompositions of a given tensor. These spaces are themselves algebraic varieties. P3. Real ranks: the rank of a tensor depends on the field of coefficients. Over the real numbers, we know very little compared to the case of complex numbers. P4. Algebraic statistics: Several statistical models can be regarded as algebraic varieties parameterised by tensors admitting certain decompositions.