Topics in Algebraic Spline Geometry

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The aim of the project is to explore algebraic geometry tools for the study of approximation methods based on piecewise polynomial functions, also known as splines. Splines are a powerful tool widely used in different challenges arising in computational mathematics, such as the numerical approximation of partial differential equations (the finite element method), the parametrization of shapes in geometric modelling and computer-aided geometric design, and data science. This project will be oriented towards the theoretical aspects of the topic, we will focus on algebraic aspects of splines, their ring structure, and connections with the theory of toric varieties.