Otis Chodosh of Stanford University presents a research talk given at the Institute for Advanced Study as part of a workshop honoring Jim Simons. The talk takes up Bernstein's problem, the classical question of whether a minimal graph defined over all of space must be an affine plane, and traces how the answer changes as dimension grows. Chodosh connects this to the broader theory of regularity for area-minimizing hypersurfaces, the study of where such surfaces can fail to be smooth and how singular sets behave in high dimensions. He walks through the geometric measure theory machinery used to control these singularities and discusses recent progress, including work pushing past the dimensions where the classical Bernstein theorem holds. The talk assumes familiarity with minimal surface theory and is aimed at researchers, giving a condensed account of where the field stands on a problem that has occupied geometers since the 1910s.