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> I really would have liked a discussion of why people assumed the conjecture must be true, given that the only analogy presented was obviously not true.

I'm trained in the field (on the math side, actually at the same institution as Vern Paulsen quoted in the article), and I never got the impression that the answer was universally assumed to be true. One of the major consequences of a positive answer would have been that every countable discrete group is hyperlinear, which would have violated a well-known heuristic principle of Gromov that every property of all finitely generated groups is either false or trivial.

The frustrating aspect about the refutation via quantum information theory is that there is no obvious way to transfer the insight to also refute the consequences of a positive answer to the problem.

Cwzwarich could you explain further what the Gromov heuristics is about? or where could I read more about it? Thank you

The analogy holds. For a 2x2 matrix the error bound is obviously quite large, but you still know it exists. Physics do not allow you to simply modify the temperature of a point. Thermodynamics prevent that.

Now the interesting question is, does the conjecture follow from the model itself? Apparently not. But I do neither understand the model nor the proof.

> Physics do not allow you to simply modify the temperature of a point. Thermodynamics prevent that.

gotcha - in terms of the analogy, the temperature distribution cannot be any arbitrary distribution with a finite average (i.e. a finite L2 norm) --- including those pathological cases e.g. the temperature is 0 everywhere except at one point x where it takes the value 1 ---- no, the temperature distribution has to have some minimum smoothness structure, meaning that if you adjust the temperature at a single point, you have to also locally adjust the temperature of neighbouring points enough for it to make a difference to the overall global average temperature over the entire space.

for intuition, the animation on the wikipedia "heat equation" page is a good start: https://en.wikipedia.org/wiki/Heat_equation

When you plug initial conditions into the heat equation, over time it evolves to be smoother.

But, according to the article, the Connes conjecture applies to all infinite matrices. There's nothing strange about modifying the value of one entry in a matrix.

Quite a nice result. Along the lines of your point 2., it's not the first time in recent years a problem in operator algebras was resolved by the computer science community using unexpected methods: https://www.quantamagazine.org/computer-scientists-solve-kad.... Perhaps looking at these problems from a more concrete angle is the way to go, especially for a problem like disproving the Connes conjecture.

As this is referring to photons I'm reminded of a concept that comes from the use of Feynman diagrams to model interactions. I think this answers your question.

If you're calculating the probability that a photon goes from A to B you can calculate the probability (or more technically, the amplitude) that it goes straight there (this is the crude approximation), but you also have to account for the case where it went a bit off the way, hit another photon and then arrived at B (a slightly less crude approximation), and you can account for the path where the photon interacted with two other photons for a more accurate approximation, and with three other photons for an even more accurate approximation.

People stop the calculation from being infinitely complex because it appears that the probability of the cases with many interactions is incredibly unlikely. I guess the conjecture here was that if you have some error bound for the N interactions case, then the N+1 interactions case would have lower error. And that's been proven to not always be true.

I may be way off the mark, my knowledge of QED comes purely from watching Feynman and Susskind on YouTube.

If I understand the article correctly, the conjecture in fact holds for all of the kinds of infinite matrices we know about and had tested it with. So either the example in the article is not one of the known examples of the conjecture working (which is not likely, given that the article says we are yet to find the matrices for which this doesn't work), or there is something else going on with the terms used.

I think what they are trying to describe in laymans terms is the delta-epsilon definition of uniform continuity. That for any desired error bound you can reach it with a fine enough mesh.