Approximating shapes with algebraic spline geometry

该项目的总体目标是探索交换代数、代数几何和组合学的工具，以研究基于分段多项式函数（也称为样条）的近似方法。该项目将侧重于多元样条函数的代数方面，特别是那些支持三角剖分和四面体分区。为此，我们将结合同调代数、脂肪点方案和协同的元素，以及刚性理论和符号计算的方法，包括计算代数几何和Groebner基。在整个项目中，学生将补充他们在代数和/或组合与几何方面的背景知识，并了解研究多元样条函数的不同方法。该项目将以学生对该主题的理论或计算方面的兴趣为导向。

The overarching aim of the project is to explore tools from commutative algebra, algebraic geometry, and combinatorics to study approximation methods based on piecewise polynomial functions, also known as splines. The project will focus on algebraic aspects of multivariate spline functions, particularly those supported on triangulations and tetrahedral partitions. For this, we will combine elements from homological algebra, fat points schemes, and syzygies, along with methods from rigidity theory and symbolic computation, including computational algebraic geometry and Groebner bases.Throughout the project, the student will complement their background in algebra, and/or combinatorics & geometry and develop an understanding of the different methods to study multivariate spline functions. The project will be oriented towards the student's interest on theoretical or computational aspects of the topic.