
Mathematicians try to understand how CS proof solves Connes embedding conjecture - dnetesn
http://abstractions.nautil.us/article/538/graced-with-knowledge-mathematicians-seek-to-understand
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thaumasiotes
It could just be me, but this article:

1\. Describes a simplified analogy to the Connes conjecture. The analogical
conjecture is obviously false. ("The average temperature over some portion of
the earth's surface must necessarily approximate the actual temperature at
each point within the portion, with bounded error." That can't be true,
because while modifying a single point affects the average temperature, you
can modify a set of multiple points, including any arbitrary increase or
decrease to one of those points, without affecting the average at all. Thus,
there is no bound on the deviation between the temperature at an individual
point and the average temperature over every point.)

2\. Goes on at length about how everyone just assumed the Connes conjecture
would turn out to be true, and they've all been blindsided by a proof that it
isn't.

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.

~~~
choeger
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.

~~~
shoo
> 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](https://en.wikipedia.org/wiki/Heat_equation)

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

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fjfaase
The paper 'MIP* =RE'
[https://arxiv.org/abs/2001.04383](https://arxiv.org/abs/2001.04383), where
the article is referring to, has already been discussed here extensively. It
is also being studied by many people, because it is a very important and
surprising result. I guess that Connes conjecture is but one of the many
things that is affected by the MIP* =RE result.

~~~
chrispeel
[https://news.ycombinator.com/item?id=22083935](https://news.ycombinator.com/item?id=22083935)

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aklein
I truly enjoy all of Kevin Hartnett’s articles. He’s a great science writer.
He shines a light on some really impenetrable ideas.

