Approximating shapes with algebraic spline geometry

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.