Recent Developments on Geometric Measure Theory and its Applications

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This award provides partial participant support of the conference "Recent Developments on Geometric Measure Theory and its Applications", to be held at Rice University (Houston, Texas) on 19-21 March 2020. A common tendency of both nature and human engineering is to seek maximum efficiency in solving a problem. Nature will design a leaf to catch the sun and transport nutrients, subject to the constraining influence of the location of the plant; a soap film will use the least amount of material to span a boundary; humans will design a road system to most efficiently move people and goods from place to place. The study of what the optimal shapes are for natural and artificial design problems has been a focus of increasingly sophisticated study by mathematicians for centuries, involving techniques from the foundations of calculus, geometry and partial differential equations. In the past few years, there have been breakthroughs in a number of disparate areas of this subject, but few conferences in the United States that bring together experts from across the range of the subject to meet and exchange perspectives in a single meeting. This conference aims for such a mixture of ideas.We list a number of areas where there has been deep recent progress. First, there have been important developments in the basics of geometric measure theory in general metric spaces, both finite and infinite dimensional. There has also been enormous progress in geometrical variational problems, both in a number of areas of regularity theory and in minimax theory for minimal hypersurfaces. We also see deep results in the structure of nodal sets, mean curvature flows and in harmonic measure. Further details are available at https://math.rice.edu/NewsEvents/Conferences/BobHardtGMTConference/index.html.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.