Executive Summary : | The project focuses on the study of Hilbert functions and their applications, which are central to various branches of mathematics, including Commutative algebra, Algebraic Geometry, Combinatorics, and Topology. The Hilbert function, introduced by Hilbert in 1890, has been extensively researched in various areas, including algebraic geometry, singularity theory, commutative algebra, topology, and combinatorics. However, several central problems remain open, including the classification of Hilbert functions of an affine or projective variety whose defining ideal is Gorenstein or complete intersection. This classical problem has been extensively studied, starting with Macaulay's work in 1916. The project also aims to apply the Hilbert function to tackle challenging problems in different branches of mathematics, particularly the Waring problem for forms, which has applications in areas such as statistics, biology, quantum information theory, signal processing, data mining, and machine learning. The project is motivated by the significant interest in international research on Hilbert functions and recent works with L. Brustenga J. Jelisiejew and M. E. Rossi. |