Algebraic spline geometry: towards algorithmic shape representation

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The increased demand for 3D visualization and simulation software in medicine, additive manufacturing, architectural design, and mechanical engineering, among many other areas, gives rise to new mathematical challenges in applied geometry and approximation theory. At the same time, a new paradigm emerges with the potential use of Machine Learning in Computer-Aided Design and Manufacturing (CAD/CAM) to improve the modelling experience, allowing users to anticipate and repair errors in real time. In this context, understanding the mathematical foundations behind the storage, manipulation and analysis of complex shapes is essential for the development of more accurate and efficient computational methods.This 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. The objective of this project is to develop novel representation techniques for complex shapes by exploiting the ubiquity of splines in algebraic geometry and approximation theory. Splines have been traditionally studied within the realm of numerical analysis and computational mathematics. Instead, the originality of this project resides in proposing an integrated approach to mathematical questions lying at the heart of splines by using methods stemming from algebra, geometry, topology and combinatorics.