Anagrafe della ricerca

Multilinear maps are ubiquitous in mathematics, especially in their symmetric version representing both homogeneous polynomials and symmetric tensors. The vastness of interests that they open penetrates the depths of algebraic, geometric and numerical research which by their nature take roots in classical algebraic geometry and find an outlet in highly relevant open problems such as complexity theory. In this project we aim to study the geometry of these spaces describing multilinear phenomena, with a focus on secant varieties and the connected concept of rank harmonizing the whole project around one of the most important unsolved problems in complexity theory like the complexity of Matrix Multiplication (MaMu). The interest in these multilinear phenomena is increasing due to their interdisciplinary nature, which is spread over several fields of Mathematics (Multilinear Algebra, Commutative Algebra, Algebraic Geometry, Combinatorics, Numerical Analysis). Tensors are nowadays widely used in several emerging technologies: Signal processing, Machine learning, Algebraic statistics, Game Theory, Complexity. We list here the four main topics whose related specific targets will be detailed in the next sections: 1. Complexity of Matrix Multiplication 2. Identifiability 3. Zero-dimensional schemes and cactus varieties 4. Critical points of distance function and low rank optimization Common tools in the four above items are vector bundles and projective geometry. We also employ symbolic computational methods joint with methods from Numerical Algebraic Geometry. The goal of this project is to obtain financial support for the continuation of the research in the algebraic and geometric aspects of tensors theory, as well as the production of algorithms and the consequent software for tensor analysis. We plan to use the funding to: - support promising young researchers to undertake the main tasks of the present project, by providing them with post-doc grants in the Universities involved in the project; - enhancing the collaboration among researchers in the group, by supporting their mobility and by organizing meetings among them; - support the worldwide participation of members of the group to meetings of top level on the topic and organize similar events in Italy; - invite outstanding foreign researchers in the field, for delivering advanced courses in the topics of the project and for performing joint research with members of the group.