Harmonic Maps into Singular Spaces and Applications

Maps help people understand complicated places by displaying distances, shapes, and spatial relationships in a clear form. Mathematicians also use maps to compare geometric spaces—settings in which concepts such as distance, shape, and motion can be studied. Some geometric spaces resemble familiar smooth surfaces, while others have sharp, branching, or irregular features for which the standard tools of calculus do not apply directly. Harmonic maps are especially useful for making such comparisons because they are selected by an energy principle that measures how much a map stretches or distorts a space. In this project, the principal investigator (PI) will develop new methods for studying harmonic maps whose destination spaces may not be smooth. These methods will deepen understanding of geometric structures that arise throughout mathematics and in mathematical descriptions of physical phenomena. This foundational work serves the national interest by promoting the progress of science, creating broadly applicable mathematical tools, and strengthening the nation’s scientific workforce. The project will also support the education and professional development of graduate students, postdoctoral researchers, and other early-career mathematicians. The PI will disseminate the results through publications, lectures, conferences, and mentoring activities. Harmonic maps are central tools in geometric analysis because they encode geometric, topological, and algebraic information about spaces. The project will study harmonic maps into complete metric spaces with non-positive curvature in the sense of Alexandrov, commonly called NPC metric spaces, with particular emphasis on targets that lack a smooth structure. Its principal goals are to establish new regularity results for such maps, develop methods that allow differential-geometric techniques to be used in non-smooth settings, and apply the resulting theory to problems in geometric group theory, complex geometry, and algebraic geometry. The work will combine variational methods for energy-minimizing maps, metric geometry, regularity analysis, and geometric arguments adapted to non-smooth targets. By extending harmonic-map methods beyond the smooth setting, the project seeks to address longstanding mathematical problems, create new tools for studying complex geometric and algebraic structures, strengthen connections among distinct areas of mathematics, and open directions for future research. 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.