

Mistake found in invariant subspace proof - ColinWright
http://cafematematico.com/2013/02/05/statement-from-cowen-and-gallardo/

======
ot
Some context: the Invariant Subspace Problem is an important open problem in
functional analysis [1]. There is a MathOverflow thread discussing its
significance [2].

For those not familiar with functional analysis, it is basically a
generalization of basic linear algebra fact that, in complex finite-
dimensional vector spaces, every matrix (or to be more precise, every linear
operator) has at least one eigenvector.

Cowen and Gallardo announced [3] that they proved the theorem in December, but
apparently the proof is wrong, so the problem is still open.

[1] <http://en.wikipedia.org/wiki/Invariant_subspace_problem>

[2] [http://mathoverflow.net/questions/48908/is-the-invariant-
sub...](http://mathoverflow.net/questions/48908/is-the-invariant-subspace-
problem-interesting)

[3] [http://aperiodical.com/2013/01/the-invariant-subspace-
proble...](http://aperiodical.com/2013/01/the-invariant-subspace-problem-
solved-for-hilbert-space/)

~~~
user24
> For those not familiar with functional analysis, it is basically a
> generalization of basic linear algebra fact that, in complex finite-
> dimensional vector spaces, every matrix (or to be more precise, every linear
> operator) has at least one eigenvector.

Oh, thanks for the clarification.

~~~
ot
I sense some sarcasm :) Luckily these things have nice geometrical
interpretations, let me try to clarify this better, hoping that I can
stimulate some curiosity on the subject.

Imagine a _linear_ transformation in the 3-dimensional space; that means that
rotations, "flips", and scalings are allowed. For example take a rotation:
points that are in the axis of rotation remain on that line. Likewise, think
of the plane _orthogonal_ to the axis of rotation, then points that are on
that plane rotate, but they stay on that plane. The axis of rotation and its
orthogonal plane are both _invariant subspaces_.

In a 3-dimensional space, and in general in every odd-dimensional space, every
linear transformation has an invariant subspace. This is not true for even-
dimensional spaces, such as the 2d plane: a 45-degree rotation for example has
no invariant subspace (except for {0} and the whole space, of course).
However, if you allow the coordinates to be _complex numbers_ , instead of
real, then there always is an invariant subspace, although it is harder to
visualize.

To express this in linear algebra terms, every complex matrix has an
eigenvector. This is a fundamental fact that has applications pretty much
everywhere in mathematics.

The Invariant Subspace problem generalizes this statement to spaces that are
infinite-dimensional. Infinite-dimensional spaces are infinitely messy, so we
restrict our attention to spaces that have some additional structure; for
example Hilbert spaces, those addressed in the paper, are those where a
formula very similar to the Pythagorean theorem is true.

~~~
user24
Thanks for your detailed reply.

It's a lot to digest. As I explained in another reply, my mathematics is very
poor indeed.

> Likewise, think of the plane orthogonal to the axis of rotation, then points
> that are on that plane rotate, but they stay on that plane.

Why only the orthogonal plane, I don't see how any set of points behaves
differently to any other. For example the stars could be said to rotate around
the earth (I know, but pretend). They don't move in relation one another
though. They only change relative to the earth. I think I'm grossly
misunderstanding. Does 'rotation' have a special meaning in this context?

> 45-degree rotation for example has no invariant subspace

because you can rotate points on a line 45 degrees and the only one that stays
the same is 0,0, which you call {0}?

So.. someone tried to prove that for n dimenional (odd?) spaces with
n<=infinity, there is always some subspace which doesn't differ when you apply
complex transformations to it? But was wrong.

~~~
nickpinkston
To get better intuition without a math degree I'd recommend: Concepts of
Modern Mathematics which only requires elementary algebra, but will give you
great intro treatments of everything from groups to number theory.

[http://www.amazon.com/Concepts-Modern-Mathematics-Ian-
Stewar...](http://www.amazon.com/Concepts-Modern-Mathematics-Ian-
Stewart/dp/0486284247/ref=pd_sim_b_2)

------
j2kun
This is one of the worst things that can happen to a paper of yours, the only
worse thing being that it is published before you have to retract it.

~~~
gfodor
I'd say it'd be worse if someone else found the bug.

------
JackFr
I find it interesting that the phrase was 'a gap was discovered', rather than
'we discovered a gap' or 'Prof. X discovered a gap.'

Assuming they are able to prove it, would that rate a footnote 'We are
indebted to Prof. X for pointing out ....'?

And also is there a protocol or custom about how long other guys will give
them to try to fix it before they try themselves?

~~~
user24
That's likely just the academic register. I wrote my dissertation saying "we
found that..." when really it should have been "I found that...".

Getting the academic tone right in essays/thesis/papers is important and
indeed becomes second nature after a while.

------
speeder
Impressive that they admitted the mistake instead of going on full defense and
blame something else.

~~~
jgrahamc
Who could they have blamed? It's a proof: either it holds up to examination or
it doesn't.

~~~
speeder
Some academics enter in denial mode and blabber bullshit.

Others just invent convoluted ways to say that whoever pointed them wrong are
wrong.

Seemly in their case they found out they were wrong themselves and retracted
their claims quickly.

~~~
jerf
The more objective the discipline, the less you see that, because
metaphorically, the light is too bright to hide. Mathematics is, arguably, the
most objective discipline there is, so I don't think you see much of this.
Particle physics is the most concrete of the real physics, so you see
admirable scientific discipline there too, for instance, with the recent FTL
neutrinos.

As you get fuzzier and fuzzier, you get more and more blabbering, until, ahem,
eventually the "science" consists of little more than blabber; I'll leave it
as an exercise to the reader as to where to draw that line.

<http://xkcd.com/451/>

~~~
knowtheory
Er. Let's try this a different way.

Kant drew two distinctions, Analytic vs Synthetic
([http://en.wikipedia.org/wiki/Analytic%E2%80%93synthetic_dist...](http://en.wikipedia.org/wiki/Analytic%E2%80%93synthetic_distinction)
) and A Priori vs A Posteriori
(<http://en.wikipedia.org/wiki/A_priori_and_a_posteriori> ).

By Kant's reckoning. Math is Analytic and A Priori, meaning that it is neither
dependent on facts about the world, nor does the exploration of math require
knowledge of the world to interrogate.

You start with some axioms, and you just go from there.

Fields like physics are synthetic and a posteriori, and the search for a grand
unified theory which pull together the fundamental forces we know about are
intellectual endeavors that are _about_ the universe, and contingent upon the
nature of the universe. If the universe were different, the facts and
conclusions we would arrive at would be different.

None of this has to do with objectivity per se. Physics is objective(ish), but
it's not an analytical field (as Kant defines it).

And you can claim that biology or linguistics or sociology aren't "science".
But biology was still a field of scientific inquiry before Watson & Crick, or
Darwin (although a less rigorous or well developed field). We have ways to
know the world, and we should avail ourselves of them, and endeavor to improve
their rigor.

So, part of what i'm saying is that it'd be nice if people would stop
dismissing other fields as "not science" :P

~~~
neutronicus
> So, part of what i'm saying is that it'd be nice if people would stop
> dismissing other fields as "not science" :P

'Twould also be nice if people would stop trying to usurp the authority of
physics in the public policy domain.

~~~
knowtheory
These goals, they are not incompatible ;)

Also, I'm curious as to which usurpations you are alluding.

~~~
neutronicus
Social scientists try very hard to capitalize on the epistemological clout
associated with the word "science", epistemological clout derived mostly from
the successes of physics, chemistry, and biology in the
industrial/medical/policy arena.

The "not science" squabbles are an inevitable result of
physics/chemistry/biology not wanting their brand diluted, and the social
sciences wanting to, IMO, inflate the value of their own brand.

