Higher Multiplicity Singularities in Geometric Variational Problems

This project will investigate the fundamental mathematical rules that govern how certain physical shapes naturally form, balance, and change over time. The principal investigator will study the equilibrium configurations of physical systems, such as the behavior of soap films, the boundaries between different physical states in phase transitions (e.g., melting of ice and freezing of water), and how surfaces evolve under their own surface tension. Experiments suggest that such systems develop "singularities"—sharp creases, crossing points, or pinch-offs—which present significant mathematical challenges. This project seeks to understand the size and structure of these complex singular configurations. By advancing our understanding of these geometric phenomena, this project promotes the progress of science and provides foundational knowledge applicable to pure mathematics and physical modeling. Furthermore, the project serves the national interest and advances national prosperity and welfare by actively training undergraduate students, graduate students, and postdoctoral fellows. Finally, the project explicitly aligns with current national priorities in Artificial Intelligence (AI) by exploring and integrating emerging AI technologies into teaching and training tools. The project will establish foundational regularity theories for singularities arising from high multiplicity in three specific geometric variational problems: minimal surfaces, the two-phase free boundary problem, and mean curvature flow. The primary goals are to estimate the optimal dimension bounds for the singular sets of stationary integral varifolds and the branching sets of two-phase free boundary problems, as well as to characterize the first singular time of mean curvature flows originating from smooth initial data. To achieve these goals, the investigator will employ several cutting-edge techniques and develop new tools. Through these methods, the project will contribute new techniques to geometric measure theory and partial differential equations, providing crucial intermediate steps to resolve decades-old conjectures regarding high multiplicity singularities. 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.