Abstract:
For decades, the prevailing expectation was that area-minimizing submanifolds are generically smooth. I demonstrate the opposite: on almost any manifold, these minimizers exhibit fractal singular sets and are not generically smooth in almost all cases. These results disprove the intuition of regularity and resolve foundational conjectures by Almgren, Morgan, White, and Yau. I conclude by introducing a unifying framework that surprisingly links these geometric singularities to the Hasse principle from number theory.

Bio:

Zhenhua Liu is currently a research associate at Harvard University. He obtained his bachelor's degree from Duke University in 2019 and Ph.D. from Princeton University in 2026. His research focuses on geometric measure theory and area-minimizing submanifolds.