Whitney's Extension Problems

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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.