What this is yet another demonstration of how being immensely accomplished in one field says absolutely nothing about what one may be able to contribute to another.
Yes, as someone who prefers developing the 'correct' structure over 'merely' proving theorems, this is good for me in the short term. However, the writing I fear is on the wall for my medium and long term utility.
Speaking as a mathematician, it does seem like we're a bit fucked as a community. Anything that is at all accessible to currently existing methods and mathematical infrastructure is probably going to fall to the frontier models of today, and at this rate of progress it's likely that, already by next year, we'll see new infrastructure being put into place by AI, giving us a world in which a few designated interpreters of the oracle get to 'do' mathematics, while it withers on the vine as an avenue for the exploration of human meaning.
This is nonsensical: Properness of the map is equivalent to its being an isomorphism (quick proof: Jacobian invertible implies that the map is etale, and properness would imply that it is finite etale, but affine space doesn't admit non-trivial finite etale covers), so the lack of properness is just another way of verifying that this is indeed a counterexample.
This is of course an oversimplification by graduate students trying to reach a broad audience, but one could argue that it in fact is. Langlands himself viewed his program as part of an attempt to achieve non-abelian reciprocity laws. See for instance: https://www.phys.ens.psl.eu/~kashani/slides_chenevier.pdf
>The situation with human mathematicians is not much different. Eg, Wiles original proof of Fermat's Last Theorem contained errors found by reviewers, which he later repaired.
In fact, it was Wiles himself who realized there was an error.
I don't know the numbers, but you and the parent are saying different things. The US could still be supplying 2/3s of Israel's weapons while only subsidising about 10% of that supply.
”For example, it allows NSF to make awards to nontraditional recipients such as a limited partnership or a venture capital firm, some of which might have been created solely for the purpose of receiving the NSF award. It also allows NSF to make additional awards without the need to review a new application.”
FTA: The public comment period closes approximately July 13, 2026 (45 days from May 29 publication). Comments must be submitted to regulations.gov, Docket OMB-2026-0034.
This kind of comment might have been borderline reasonable before 2025 (though I would have found it laughable even then), but the evidence of the last 18 months shows that one side is in fact evil, corrupt and venal in entirely unprecedented ways.
Covariance and contravariance are mathematical notions, and have to do with whether each multiplicative constituent or the tensor is a vector in your given vector space (covariant) or a linear functional on this space (contravariant). There is no inherent physical meaning to either concept.
Hard to read this and conclude that there can be any actual way to produce admissible evidence. Can you give me an example of what conversations you can use as evidence if anything involving him or his advisers is off limits?
Geometric Langlands, while inspired by questions Weil was interested in, is actually answering somewhat different questions, which are not arithmetic in nature: hence the 'geometric' moniker. The actual Langlands program, which deals with number fields and hence with questions of an arithmetic flavor (meaning solving equations over the rational numbers rather than the real or complex numbers) is still very much unexplored in its full generality.
Edit: it appears that I might have spoken too soon. At the link below is a paper by Sam Rankin establishing some consequences for arithmetic questions, though over function fields (such as those formed by rational functions over finite fields).