Topics in Algebraic Spline Geometry

该项目的目的是探索用于研究基于分段多项式函数（也称为样条）的逼近方法的代数几何工具。样条函数是一种强大的工具，广泛用于计算数学中出现的不同挑战，例如偏微分方程的数值逼近（有限元法），几何建模和计算机辅助几何设计中形状的参数化以及数据科学。这个项目将面向该主题的理论方面，我们将专注于样条的代数方面，它们的环结构，以及与复曲面簇理论的联系。

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.