Research on Neutrinos Leads to Breakthrough in Basic Mathematics

[MJ12 Commando]

Neutrinos Lead to Unexpected Discovery in Basic Math | Quanta Magazine

Three physicists stumbled across an unexpected relationship between some of the most ubiquitous objects in math.

www.quantamagazine.org

Article:

After breakfast one morning in August, the mathematician Terence Tao opened an email from three physicists he didn't know. The trio explained that they'd stumbled across a simple formula that, if true, established an unexpected relationship between some of the most basic and important objects in linear algebra.

The formula "looked too good to be true," said Tao, who is a professor at the University of California, Los Angeles, a Fields medalist, and one of the world's leading mathematicians. "Something this short and simple — it should have been in textbooks already," he said. "So my first thought was, no, this can't be true."

Then he thought about it some more.

Eigenvectors from eigenvalues

Peter Denton, Stephen Parke, Xining Zhang, and I have just uploaded to the arXiv the short unpublished note “Eigenvectors from eigenvalues”. This note gives two proofs of a general eige…

terrytao.wordpress.com

No applications yet, but since eigenvectors and eigenvalues are pretty important in a whole lot of fields, this might actually be a pretty significant advance.

 

[xa na xa]

The youths of today are so spoilt!

Back in my days, you could only get eigenvalues out of eigenvectors!

And you could only input matrices of uphill coordinates!

 

[Dark Lord Bob]

From the article:

"To our surprise, he replied in under two hours saying he'd never seen this before," Parke said. Tao's reply also included three independent proofs of the identity.

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One of my goals, when it comes to mathematical maturity, is to be able to comfortably read Tao's blog. He writes well in the few parts I can understand.

 

[linkhyrule5]

Eigenvectors from eigenvalues

Peter Denton, Stephen Parke, Xining Zhang, and I have just uploaded to the arXiv the short unpublished note “Eigenvectors from eigenvalues”. This note gives two proofs of a general eige…

terrytao.wordpress.com

No applications yet, but since eigenvectors and eigenvalues are pretty important in a whole lot of fields, this might actually be a pretty significant advance.

Click to shrink...

Man, what I wanna know is, does this generalize to infinite-dimensional Hilbert spaces?

Because geez, but the number of quantum systems we can get the energies for but not the actual exact eigenfunctions is stupid huge. If we can actually get some more exact solutions that would be beautiful.

 

[Roadie]

I wish I remembered enough math to actually understand the implications of this.

 

[Shane_357]

Okay that title is completely misleading. Eigen-whatevers are not basic math.

 

[confusopoly]

Okay that title is completely misleading. Eigen-whatevers are not basic math.

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I think this means basic in the sense of "a base for a lot of different things", not in the sense of simple and easy to understand.

 

[13th Bee]

Okay that title is completely misleading. Eigen-whatevers are not basic math.

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They really are pretty basic. Linear algebra generally loses out to calculus for which gets taught first, but it's a simpler subject in my opinion. Trigonometry courses are usually harder than linear algebra ones, and those are standard high school subjects.

The name and formal math definitions can be pretty confusing, but a formal mathematical definition of basic addition can also be headache inducing. That's just what formal math definitions are like for most non-mathematicians. The actual concepts behind eigenvalues and eigenvectors aren't too bad once you get through the math jargon, and they really are fundamental to a whole lot of stuff.

 

[Quadhelix]

Okay that title is completely misleading. Eigen-whatevers are not basic math.

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On top of @13th Fleet's overall point, I think it's worth emphasizing that, even if most non-mathematicians and non-physicists may have never heard of matrix operations, eigenvalues, and eigenvectors, those are extremely basic things for career mathematicians and career physicists -- and that's kind of the point here. The people who made and confirmed this discovery had likely been using eigenvectors and eigenvalues since their first year of undergrad. Moreover, every mathematician and physicist in the world learned about eigenvectors as part of their basic maths education.

So what's particularly newsworthy about this is that it's as unexpected as discovering a new species of elephant in your backyard.

 

[MJ12 Commando]

They really are pretty basic. Linear algebra generally loses out to calculus for which gets taught first, but it's a simpler subject in my opinion. Trigonometry courses are usually harder than linear algebra ones, and those are standard high school subjects.

The name and formal math definitions can be pretty confusing, but a formal mathematical definition of basic addition can also be headache inducing. That's just what formal math definitions are like for most non-mathematicians. The actual concepts behind eigenvalues and eigenvectors aren't too bad once you get through the math jargon, and they really are fundamental to a whole lot of stuff.

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I remember just enough of linear algebra that I can say with some confidence that having to actually prove the functions behind basic arithmetic is much, much harder than just memorizing them.

 

[ctulhuslp]

I remember just enough of linear algebra that I can say with some confidence that having to actually prove the functions behind basic arithmetic is much, much harder than just memorizing them.

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Well, yes, but it's
a) part of uni's curriculum in such things anyway, it's not like they let you just memorize
b) useful for understanding and internalizing hows and whys of theory

 

[Nyvis]

They really are pretty basic. Linear algebra generally loses out to calculus for which gets taught first, but it's a simpler subject in my opinion. Trigonometry courses are usually harder than linear algebra ones, and those are standard high school subjects.

The name and formal math definitions can be pretty confusing, but a formal mathematical definition of basic addition can also be headache inducing. That's just what formal math definitions are like for most non-mathematicians. The actual concepts behind eigenvalues and eigenvectors aren't too bad once you get through the math jargon, and they really are fundamental to a whole lot of stuff.

Click to shrink...

The problem with linear algebra, from what I remember from the distant times when I learned math, is that it's a bunch of new vocabulary and objects people aren't familiar with, while stuff like calculus and trigonometry get introduced along with more practical concepts in physics. Trigonometry is a big pain but it's usually possible to do a bit of drawing to relate to what's on the page. And calculus is directly involved in pretty basic physics so you have to go there. Linear algebra usually doesn't come until you start physics of systems, solids and liquids, rather than the usual "points with mass" of high school physics.

 

[Axslashel]

The problem with linear algebra, from what I remember from the distant times when I learned math, is that it's a bunch of new vocabulary and objects people aren't familiar with, while stuff like calculus and trigonometry get introduced along with more practical concepts in physics. Trigonometry is a big pain but it's usually possible to do a bit of drawing to relate to what's on the page. And calculus is directly involved in pretty basic physics so you have to go there. Linear algebra usually doesn't come until you start physics of systems, solids and liquids, rather than the usual "points with mass" of high school physics.

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I would say that knowing how to calculate with vectors is pretty fundamental for basic physics. And writing a system of equations into a matrix is very useful.

 

[ctulhuslp]

The problem with linear algebra, from what I remember from the distant times when I learned math, is that it's a bunch of new vocabulary and objects people aren't familiar with, while stuff like calculus and trigonometry get introduced along with more practical concepts in physics. Trigonometry is a big pain but it's usually possible to do a bit of drawing to relate to what's on the page. And calculus is directly involved in pretty basic physics so you have to go there. Linear algebra usually doesn't come until you start physics of systems, solids and liquids, rather than the usual "points with mass" of high school physics.

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We had linalg at first uni course at least; and I am not sure, but I think we had some matrixes/vectors and whatnot in school too, along with natural loads of trig and basics of calculus.

And, like...vectors are easily conceptualized as extension of vectors in 2/3D (coordinates); matrices are...well, yeah, example is systems of equations.

It's pretty intuitive, at the start; at least when compared to something hideous like functional analysis or topologies.

 

[Nyvis]

I would say that knowing how to calculate with vectors is pretty fundamental for basic physics. And writing a system of equations into a matrix is very useful.

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Very basic knowledge about vectors, yeah. I didn't hear anything about matrixes until I left high school.

We had linalg at first uni course at least; and I am not sure, but I think we had some matrixes/vectors and whatnot in school too, along with natural loads of trig and basics of calculus.

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Our first post high school class was about topological spaces. I think the goal was to set the tone and weed out the weak by starting with the most abstract and theoretical stuff possible

 

[Vyslanté]

I didn't hear anything about matrixes until I left high school.

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That has changed, actually. Of course, the price for that was "remove differential equations from the cursus", so...

(Myself, I've always found matrixes and related stuff incredibly hard to visualize, in opposition to things like analysis. Which, according to my classmates of that time, is weird as fuck )

 

[ctulhuslp]

That has changed, actually. Of course, the price for that was "remove differential equations from the cursus", so...

(Myself, I've always found matrixes and related stuff incredibly hard to visualize, in opposition to things like analysis. Which, according to my classmates of that time, is weird as fuck )

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It is weird.
Well, until matrixes become scary operator/function/functor/etc representations and then it all becomes one big blob of math.

 

[MJ12 Commando]

Well, yes, but it's
a) part of uni's curriculum in such things anyway, it's not like they let you just memorize
b) useful for understanding and internalizing hows and whys of theory

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I mean I'm not saying it's a bad thing either and I fully understand why these things exist and why they're taught. I'm just saying that these are pretty basic in terms of Math Math, rather than in terms of Everyday Math.

 

[Vorpal]

Eigenvalues and eigenvectors are very basic in the sense that anything that every first-year undergraduate is typically expected to understand is basic. The result itself shouldn't be considered quite that basic, though, although it's not particularly deep either.

Regardless, science reporting is generally complete rubbish, and this is not an exception. What this is is essentially some hype based on the fact that Terence Tao didn't know something that wasn't in the linear algebra textbooks that he read. Well, no offence to Prof Tao, who is definitely a world-class mathematician, but him not knowing some formula outside his field of specialisation means pretty much diddly-squat. Ok, to be very clear that's more of a complaint about the media promotion of this result rather than e.g. Tao's (Denton et al) paper on it, which actually does reference a lot of prior work and contextualise it more (but occurs after most of the earlier hype about the result and is to a great extent a response to people sending references to the authors).

As it happens, through the grapevine one can learn that it's actually decently well-known among people actually working in that stuff, and is itself a special case of a more general result. A few examples:

Chu et al., Inverse Eigenvalue Problems: Theory, Algorithms, and Applications (2005), p. 83-84.​

Meurant, The Lanczos and Conjugate Gradient Algorithms (2006), p. 20.​

I'll paste Meurant here because it's slightly more explicit:

We have seen previously that some components of the eigenvectors Tk​ play an important role in the Lanczos algorithm. In this section we give expressions for these components. The result can be expressed in different ways and can be found in many papers; see [135], [141], [144], [143], [160], [45]. However, it seems that one of its earliest appearances is in a paper by Thompson [189]; see also [190]. Actually, the result of Thompson is more general than just for eigenvectors of tridiagonal matrices.

Theorem 1.15. ... [the formula in question] ...

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Trawling for explicit formulations gives equation (7) in Thompson and McEnteggert (1968), a special case of Thompson's earlier more general result.

TL/DR: Some physicists derive an eigenvalue result that the needed but didn't know about; someone asks Terence Tao, who didn't know either, and the media goes crazy because apparently Tao not being omniscient is newsworthy. Neutrinos are basically irrelevant here.

 

[Zap Rowsdower]

TL/DR: Some physicists derive an eigenvalue result that the needed but didn't know about; someone asks Terence Tao, who didn't know either, and the media goes crazy because apparently Tao not being omniscient is newsworthy. Neutrinos are basically irrelevant here.

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Finally, I was wondering what the real explanation was. I'm in a 300-level linear algebra course this semester (bridge course I didn't do undergrad needed for the grad degree I'm working towards), and we literally started covering Eigenvectors and Eigenvalues just after this "Story" broke. I knew something had to be up with the story when the professor completely failed to note anything new relating the fundamentals he was teaching.

 

[Arden]

Eigenvalues and eigenvectors are very basic in the sense that anything that every first-year undergraduate is typically expected to understand is basic. The result itself shouldn't be considered quite that basic, though, although it's not particularly deep either.

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Quite so. It was part of my second-semester undergrad math exam, in fact. I am kind of sad that this is not as exciting as the thread title made it out to be, because I love math and I always wished I'd had enough talent for it to make it my major. Being alive to see a fundamental re-evaluation or new advance being made in a field like that is rare.

 

[tramp_stamp]

Finally, I was wondering what the real explanation was. I'm in a 300-level linear algebra course this semester (bridge course I didn't do undergrad needed for the grad degree I'm working towards), and we literally started covering Eigenvectors and Eigenvalues just after this "Story" broke. I knew something had to be up with the story when the professor completely failed to note anything new relating the fundamentals he was teaching.

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Linear algebra is actually pretty easy. It's just basic algebra where you're taught a less intuitive, but quicker, way of solving a system of equations and a way of finding an answer when the number of variables is more than the number of equations you have to work with.

Partial differential equations, OTOH, is where my brain breaks. I just don't get those.

 

[linkhyrule5]

Linear algebra is actually pretty easy. It's just basic algebra where you're taught a less intuitive, but quicker, way of solving a system of equations and a way of finding an answer when the number of variables is more than the number of equations you have to work with.

Partial differential equations, OTOH, is where my brain breaks. I just don't get those.

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PDEs look scary, but tbh they're mostly just elaborate setups for you to paste in various exponential functions and see if it works. (Or like, we don't really have any other way of solving most nonlinear PDEs anyway ). Practically 90% of modern physics is basically based on "Ansatz..." at this point...

 

[Derpmind]

As it happens, through the grapevine one can learn that it's actually decently well-known among people actually working in that stuff, and is itself a special case of a more general result. A few examples:

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Is this well-known among physicists using Eigenvalues, or just the mathematicians studying that field?

 

[Vorpal]

Is this well-known among physicists using Eigenvalues, or just the mathematicians studying that field?

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Anecdotally, people working in numerical methods for linear algebra. It's definitely obscure for mathematicians in general, never-mind physicists, and overly optimistic promotional material aside, is very likely to stay that way.

Man, what I wanna know is, does this generalize to infinite-dimensional Hilbert spaces? Because geez, but the number of quantum systems we can get the energies for but not the actual exact eigenfunctions is stupid huge. If we can actually get some more exact solutions that would be beautiful.

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Given that the formula involves taking the eigenvalues of the minors of a given matrix, almost certainly not in any useful capacity.