59-Year Old Mathematical Conjecture Disproved by ECE Professor

In 1967, mathematician Harold Grad conjectured that magnetic field and plasma pressure, as described by magnetohydrodynamic (MHD) equilibrium equations, have no smooth exact solutions unless the pressure is constant or there are certain symmetries. His theory was especially relevant to tokamaks and stellarators, both devices used to confine hot plasma long enough for fusion to occur.

Fast forward 59 years, and two papers, posted within a day of each other, have provided three families of counterexamples to Grad’s conjecture. Two of the families have been described in Landreman’s paper, Analytic toroidal 3D MHD equilibria and steady Euler flows with invariant surfaces. Both were found to have field lines that wrap helically around the toroidal plasma.

Particularly interesting to his discovery is that the results were found using assistance from GPT-6 Astra. This demonstrates the abilities and speed of AI in relation to the disciplines of science and mathematics.

The third family was discovered by Javier Gómez-Serrano, Lukas Liehr, and Mitchell Taylor and documented in their paper, Counterexamples to Grad's conjecture. In this case, the team of researchers demonstrated different solutions in which magnetic field lines close the short way around the torus.

The disproof of Grad’s conjecture is significant for two approaches to fusion energy. The stellarator fusion approach relies on highly non-symmetric geometry for confinement. 3D magnetohydrodynamic equilibria are also important for tokamaks, which operate with imperfect rotation symmetry.

Published October 14, 2026