The aim of this research project is to develop new methods of data analysis based on algebraic statistics, to demonstrate their effectiveness on real-world problems, and to make them available to the public in the form of software packages.

Three invariants of algebraic models lie at the heart of this initiative: the maximum likelihood degree (ML degree), the Euclidean distance degree (ED degree) and the polar degree. Recently developed within the field of theoretical research, these invariants promise to pave the way for new algebraic methods of data analysis. The project will bring this vision to fruition, expanding the underlying theoretical foundations where necessary and producing statistical tools that can be put to practical use by professionals in the field.

This research is expected to have four main impacts.

Firstly, the results obtained will enable the latest theoretical advances to be applied to real-world situations, improving understanding within the algebraic statistics community of the actual potential of these approaches and guiding future lines of research.

Secondly, the project generates new mathematical results of theoretical interest, developed specifically to meet the needs of the data applications addressed within the research.

Thirdly, it yields new insights into complex data analysis problems characterised by an algebraic structure, reinforcing the role of algebra and geometry as fundamental tools for data analysis.

Finally, the user-friendly software developed during the project enables algebraic techniques to be incorporated into the tools used by data analysts and experts across various application sectors, thereby promoting their wider dissemination and adoption.