FRG: Collaborative Research: New Challenges in Geometric Measure Theory

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In mathematics, computer science and operations research, optimization problems are ubiquitous. The questions are often formulated as follows: does there exist a best element (with regard to some criterion) from some set of available alternatives? Attempts to address this type of questions have played a fundamental role in the development of modern mathematics. In 1760 Joseph-Louis Lagrange asked whether there exists a surface with minimal area and prescribed boundary. This problem known as the Plateau problem was named after the physicist Joseph Plateau whose experiments with soap films yield a similar mathematical problem. The existence and regularity of such surfaces are part of Geometric Measure Theory (GMT). More generally, GMT combines methods of mathematical analysis with concepts from differential geometry, to develop the appropriate setting for studying critical phenomena in Partial Differential Equations and in the Calculus of Variations, often arising from optimization questions. Recent developments in the field forecast an imminent boom. It is the PIs' objective to capitalize on this extraordinary opportunity they are uniquely positioned to take advantage of. Fulfillment of their scientific goals will yield to developments that will shape the field for years to come. The vertically integrated structure of the PIs' teams ensures that this project will have a considerable impact in human resources. One of the PIs' main goals is to educate the next generation of researchers in Geometric Measure Theory.Pioneered in the work of Besicovitch in the thirties, the subject boomed in the fifties and sixties with the work on the multidimensional Plateau problem by De Giorgi, Federer, Fleming, Reifenberg and Almgren. The ideas developed in that extremely creative period have deeply influenced the further development of the theory of Partial Differential Equations and of the Calculus of Variations, with noticeable effects in Geometric Analysis and Mathematical General Relativity, and Harmonic Analysis and Potential Theory. The current project focuses on three major challenges in GMT, all of which are poised to have a significant impact in other areas of analysis. The PIs and their associates are expected to lead the efforts to address these problems. The challenges investigated in this project are: - Understanding singular sets of minimal surfaces and free boundaries;- Developing regularity and rigidity theorems for degenerate elliptic or non-smooth surface energies;- Quantifying local-to-global geometric rigidity results.These problems share common traits of: (i) they exemplify the most interesting open questions in the area; (ii) they have been the subject of striking recent developments, which increase the chances of their successful study; (iii) their resolution promises to have relevant impacts outside of GMT; (iv) requiring a broad approach which is encompassed by the expertise of the three PIs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.