It's notable that LLMs have now made substantial progress on four of the seven Millennium Prize problems, resolving one of them: Hodge, Birch-Swinnerton-Dyer, Riemann, and Navier-Stokes, which they resolved. No sign of P vs. NP or Yang-Mills existence and mass gap as far as I can tell, which is interesting.
Talking to some friends in physics this evening, most of the physics-related results that we could recognize were very mathematical, proving things rigorously where the physics community already had strong expectation. For instance, for a certain model of magnetism (the spin-1 Heisenberg chain), it was strongly expected that there is a finite energy gap between the ground state and the first excited state, but proving this rigorously was quite challenging. So while these are major results in mathematical physics, they probably don't rise to the level of a Millennium problem for the field.
It's interesting to think what a comparable breakthrough in physics might look like, since physics tends to favor things like conceptual understanding and applications over mathematical rigor. Maybe a new quantum algorithm, understanding of high-temperature superconductivity, a precise description of M theory...
Talking to some friends in physics this evening, most of the physics-related results that we could recognize were very mathematical, proving things rigorously where the physics community already had strong expectation. For instance, for a certain model of magnetism (the spin-1 Heisenberg chain), it was strongly expected that there is a finite energy gap between the ground state and the first excited state, but proving this rigorously was quite challenging. So while these are major results in mathematical physics, they probably don't rise to the level of a Millennium problem for the field.
It's interesting to think what a comparable breakthrough in physics might look like, since physics tends to favor things like conceptual understanding and applications over mathematical rigor. Maybe a new quantum algorithm, understanding of high-temperature superconductivity, a precise description of M theory...