Whitney's Extension Problems

Garving K. Luli的这个数学研究项目是建立在最近在经典光滑函数空间中扩展实值函数的成功基础上的，旨在发展在其他函数空间中扩展实值函数和向量值函数的理论。目标有两个：首先，在坚实的理论基础上，将扩展放在更一般的函数空间中，为开发具有实际重要性的高效算法奠定基础；第二，探索可拓问题与其他数学领域的联系。特别关注Sobolev空间的扩展，这是现代分析中无处不在和不可或缺的空间，对其的理解有望加深我们对其结构的认识。扩展问题与交换代数之间的丰富联系也将得到进一步的开发。这个数学研究项目的结果与调和分析、组合学、计算机科学、偏微分方程、几何分析、数值分析和代数几何有重要的联系。该项目中考虑的数学分析对于数据拟合至关重要，这是研究大型数据集的重要组成部分。数据拟合在许多科学和技术领域是必不可少的，因为大量数据的适当安排导致了许多学科的新发现。如今，数据拟合几乎完全在光滑函数的范围内进行。在这个项目中，通过更一般的函数进行插值的一个成熟的理论，无疑将推动数据分析的艺术状态，并将导致新的、微妙的、基本的发现，这些发现是当前数据拟合模型无法检测到的。项目中提出的许多问题都具有很强的跨学科色彩，它们将有助于汇集具有共同兴趣但不同背景的研究人员。

This mathematics research project by Garving K. Luli is built upon the recent successes in extending real-valued functions in classical smooth function spaces and aims to develop the theory for the extension of real-valued and vector-valued functions in other function spaces. The objectives are twofold: First, to put extensions in more general function spaces on a firm theoretical ground to set the stage for developing efficient algorithms that will be of practical importance; second, to explore connections of extension problems to other areas of mathematics. Special focus will be on extensions in Sobolev spaces, which are ubiquitous and indispensable spaces in modern analysis and the understanding of which holds the promise of deepening our knowledge of their structures. The rich connection of extension problems and commutative algebra will also be further exploited.Results from this mathematics research project have important connections to harmonic analysis, combinatorics, computer science, partial differential equations, geometric analysis, numerical analysis, and algebraic geometry. The mathematical analysis considered in this project is crucial to data fitting, which is an important component in the study of large data sets. Data fitting is essential in many areas of science and technology, as the proper arrangement of large amounts of data has led to new discoveries in many disciplines. Nowadays, data fitting is being carried out almost exclusively in the regime of smooth functions. A well-established theory in interpolation by more general functions, as carried out in this project, will undoubtedly advance the state of art in data analysis and will lead to new and subtle, fundamental discoveries that cannot be detected by current data fitting models. Many of the problems proposed in the project have a strong interdisciplinary flavor, and they will help bring together researchers with common interests but different backgrounds.