WiSh: Women in shape analysis 2021

Remote team on Algebraic and Geometric tools in CAGD

We will approach problems in approximation theory and computer-aided geometric design using techniques from applied geometry, commutative algebra and algebraic geometry. This project is meant for researchers with interests falling broadly under multivariate spline theory, interpolation, and geometric modeling. We will have an outlook towards practical problems arising in industrial and applied mathematics including computational geometry, computer-aided design, robotics, and optimization. The project focuses on the most recent advances for non-standard spline methods and their applications, and includes adaptive spline spaces, multi-patch spline constructions, splines on unstructured meshes, as well as subdivision schemes. In addition, related application algorithms in isogeometric analysis and image registration will be considered. The main objective of this project is to analyze these challenges from a different angle, by treating spline spaces from a commutative and homological algebra perspective. This methodology has led for instance, to rigorous proofs of computational conjectures on the dimension of spline spaces. In addition, the algebraic approach links aspects of approximation theory and algebraic geometry by taking advantage of the interplay between the combinatorics of the underlying mesh, and the algebraic properties of the polynomial pieces governed by the smoothness conditions.