This research project concerns the study of Algebraic Spline Geometry, a branch of mathematics focused on methods stemming from algebra, geometry and combinatorics, to approach problems arising in approximation theory, computational modelling, and data analysis. The word spline refers to one of the most used tools for shape approximation, they are mathematical representations built upon simpler pieces (usually defined by low-degree polynomials) which are glued together forming a smooth curve, or the surface of a volume.

What makes splines an appealing object for shape representation is that besides the simplicity of their construction, they are a fundamental component in the approximation of partial differential equations by the finite element method, playing a central role in novel fields such as Isogeometric Analysis and Computer Vision. Moreover, homological algebra techniques unveil fascinating connections between splines and algebraic geometry, putting spline theory at the interface between commutative algebra, geometric modelling, and numerical analysis.

Project Goal