Algebraic spline geometry: towards algorithmic shape representation

医学、增材制造、建筑设计和机械工程等许多领域对3D可视化和仿真软件的需求不断增加，这给应用几何和近似理论带来了新的数学挑战。与此同时，随着机器学习在计算机辅助设计和制造（CAD/CAM）中的潜在应用，一种新的范例出现，以改善建模体验，允许用户实时预测和修复错误。在这种情况下，理解复杂形状的存储、操作和分析背后的数学基础对于开发更准确、更有效的计算方法至关重要。该项目涉及代数样条几何的研究，这是数学的一个分支，专注于代数、几何和组合学的方法，以解决近似理论、计算建模和数据分析中出现的问题。样条指的是最常用的形状逼近工具之一，它们是建立在更简单的部分（通常由低次多项式定义）上的数学表示，这些部分粘在一起形成光滑的曲线，或一个体积的表面。样条曲线之所以成为形状表示的一个吸引人的对象，除了结构简单之外，它们还是有限元法近似偏微分方程的基本组成部分，在等几何分析和计算机视觉等新领域发挥着核心作用。此外，同调代数技术揭示了样条和代数几何之间的迷人联系，将样条理论置于交换代数、几何建模和数值分析之间的接口。该项目的目标是通过利用样条在代数几何和近似理论中的普遍性来开发复杂形状的新颖表示技术。传统上，样条是数值分析和计算数学领域的研究对象。相反，这个项目的独创性在于通过使用源自代数、几何、拓扑学和组合学的方法，提出了一种综合的方法来解决位于样条曲线核心的数学问题。

The increased demand for 3D visualization and simulation software in medicine, additive manufacturing, architectural design, and mechanical engineering, among many other areas, gives rise to new mathematical challenges in applied geometry and approximation theory. At the same time, a new paradigm emerges with the potential use of Machine Learning in Computer-Aided Design and Manufacturing (CAD/CAM) to improve the modelling experience, allowing users to anticipate and repair errors in real time. In this context, understanding the mathematical foundations behind the storage, manipulation and analysis of complex shapes is essential for the development of more accurate and efficient computational methods.This project concerns the study of Algebraic Spline Geometry, a branch of mathematics focused on methods stemming from algebra, geometry and combinatorics, to approach problems arising in approximation theory, computational modelling, and data analysis. The word spline refers to one of the most used tools for shape approximation, they are mathematical representations built upon simpler pieces (usually defined by low-degree polynomials) which are glued together forming a smooth curve, or the surface of a volume. What makes splines an appealing object for shape representation is that besides the simplicity of their construction, they are a fundamental component in the approximation of partial differential equations by the finite element method, playing a central role in novel fields such as Isogeometric Analysis and Computer Vision. Moreover, homological algebra techniques unveil fascinating connections between splines and algebraic geometry, putting spline theory at the interface between commutative algebra, geometric modelling, and numerical analysis. The objective of this project is to develop novel representation techniques for complex shapes by exploiting the ubiquity of splines in algebraic geometry and approximation theory. Splines have been traditionally studied within the realm of numerical analysis and computational mathematics. Instead, the originality of this project resides in proposing an integrated approach to mathematical questions lying at the heart of splines by using methods stemming from algebra, geometry, topology and combinatorics.