Can someone with more knowledge than me contextualize this result? It appears to be making progress on the Riemann hypothesis, but why is this particular effort getting attention on HN?
"In brief, the Riemann hypothesis is equivalent to the conjecture that Λ ≤ 0."
This result seems to imply that the De Bruijn-Newman constant (Λ) is non-negative, i.e., >=0. Thus, were one able to prove that Λ=0, then one would have also shown the Riemann Conjecture to be true? Similarly, if one were to show Λ to be not equal to 0, then the Riemann Conjecture would be false.
Again, just a quick reading. Sure others who are better/smarter can chime in!
Update: Tao expands on this point in the bottom of his blog post.
Yes so if you want to prove the Riemann hypothesis one possibility would be to similarly show some sort of contradiction if Λ > 0. This is sort of the opposite approach that Rogers and Tao have appeared to use. Of course maybe that side is much much harder (it is equivalent to the Riemann hypothesis after all).
I would presume that entirely different methods would be needed to prove that the constant can't be positive. There are many types analytical proofs in math that break down to "negative", "zero" and "positive" cases and often a couple of those are relatively easy whereas others are extremely difficult. The Calabi conjecture is an example.
So really it's hard to say. You might even say this proof made things harder. It guarantees that you have to prove the constant is 0 if you want to prove the conjecture. Before you might have hoped prove negativity.
Right, near the bottom of the paper he writes “...this result does not make the Riemann hypothesis any easier to prove, in fact it confirms the delicate nature of that hypothesis”
Hmm. So the Riemann hypothesis requires Λ to be <= 0. Now they’ve showed that Λ >= 0. Since the intersection of the two ranges is non-measurable, probabilistically the Riemann hypothesis must be false! :^)
{0} is measurable (It's a borel set) with the standard lebesgue measure and has measure 0. Of course, with some other measure it might not be measurable, but that would be a weird measure.
Been thinking for a while how such a measure would be defined, but my measure theory abilities are a bit rusty (heck, unused for 10 years already, time flies). I'd bet it's impossible (in \mbb{R})... and sadly I'll now keep thinking about it until I remember how to prove for sure it or fall asleep, thanks :D
You could identify 0 with some other point to get a topology where {0} is not a Borel set anymore.
A somewhat more useful example would be to identify x with x+1 for all x (or equivalently use the topology on R induced by the map t -> e^2πit from R to S^1), which results in a topology where only periodic functions are continuous.
For a complete measure you get the additional requirement that 0 isn't allowed to be contained in any null-set.
If we normalize the Fourier transform {{\mathcal F} f(\xi)} of a (Schwartz) function {f(x)} as {{\mathcal F} f(\xi) := \int_{\bf R} f(x) e^{-2\pi i \xi x}\ dx}, it is well known that the Gaussian {x \mapsto e^{-\pi x^2}} is its own Fourier transform.
Funny that he should mention it: just the other day, my mom and I were having a good laugh because as it turned out mom's neighbor, a Mrs. Cringleton, apparently had no idea that the Gaussian is its own Fourier transform under these circumstances. Apparently it came up over a cup of tea and bicuits they were having, and Mrs. Cringleton felt quite a bit embarrassed when my mom remarked: "But my dear, that is well-known!"
All jokes aside, Terry is writing to an audience where this is utterly elementary. The phrase "it is well known" is seldom misused. Yes, anyone who knows anything about the Fourier transform will know the Gaussian is a fixed point, or can easily verify it for themselves. It would actually look very amateurish if Terry provided a citation for that. In fact, it would look borderline cranky. Most of the cranky "proofs" of the RH begin by dedicating a whole section or two to elementary results, such as analytic continuation, followed by three or four sections of misunderstandings or utter gibberish.
Agreed. "It is well-known" should be read as "It is well-known among specialists".
On the other hand, I have some professors who abused the word "trivial" to mean "it can be proven by me". One professor spent the better half of an hour thinking about a question, only to declare that the answer was trivial. His explanation followed and filled the rest of the hour.
Remember that problems in high school typically can take 1/2h to solve, in college a couple of hours ... as a PhD typically a couple of years and as a professional researcher ... many years. So for him, a problem where he was confused 20 minutes and solved it in 10 minutes is probably "trivial".
There's a reason there's a somewhat famous joke about a professor who, when asked to explain an step he claimed to be trivial, furrowed his brow deeply, spent the best part of an hour looking at notes and scribbling on the blackboard only to declare afterwards "Yes! It is trivial!'.
(Not even specialists -- 2nd-3rd year undergraduates in engineering/math/physics!)
In that case the word 'trivial' means it doesn't require complex machinery to solve. A simple consequence of arithmetic manipulations, say, may be extremely difficult to think of yourself, but once you know it it's trivial to see the validity of. I believe that counts as 'trivial' in mathematical terminology.
> All jokes aside, Terry is writing to an audience where this is utterly elementary.
One of the things I find fascinating about the Internet is the effect it has on writing when you have so little control over who your audience is.
Every now and then, a blogger or Twitterer used to writing for some niche audience catches the larger eye of the Internet zeitgeist and all of the sudden momentarily finds themselves with a much larger, very different audience, who perceive their work entirely differently because they lack the context the author presumed.
I wonder how much of the modern trends of political correctness, trigger warnings, etc. come from this experience where you can no longer safely make any assumptions about your audience. At the same time, it's not possible to write effectively when you assume the audience could be anyone, even a potentially malicious aggressor.
It's the result of opening all the echo chambers to the world, including parts of the world who, for whatever reason or none at all, simply don't like you, who you are, what you are, or what you do. It also opens them to people who are just not going to get it, but who still feel compelled to comment and get a reaction, whatever kind of reaction they can elicit, and will not stop until they get a reaction or are forced to leave/stop participating.
So, can we recreate the good echo chambers, such as academia, but do it right this time, such that irrelevant characteristics no longer matter but the people inside of them no longer have to deal with nutballs and assholes coming in and disrupting anything?
Back when I worked at Twitter, celebs used to show up randomly at the HQ at lunchtime. I was just your average Indian immigrant data scientist, not clued into American culture.
The team I was on was trying to hire a lady for a front end eng role. I didn’t know what the lady’s name was, but I was told she was a very promising candidate, and we were all looking forward to the day when she joined because we had a ton of frontend work.
Ok, so at lunch, my colleague said out aloud - Hey Katy Perry is in the lunchroom!
Everyone got excited. So I said, Oh is that the new front end engineer ?
The whole room exploded in laughter. I thought I must have made a big social gaffe. My mind quickly raced...,oh maybe she must be a celebrity of some kind. Perry Perry...which Perry do I know...So previous night I was watching some show on TBS and suddenly a Perry name flashed into my brain, so I instantly said - Oh she must be the daughter of Tyler Perry!
The room just doubled up in laughter. Finally I did a wiki lookup and found out who she was. But for days, people would look at me, laugh and say he’s the Katy Perry Tyler Perry guy ha ha ha.
Finally I was so embarrassed and pissed off I asked them - Do you know the difference between between Jacobian and a Hessian ? Nobody knew. Well, I said triumphantly, I might not know your katy perry from tyler perry, but atleast I know what I need to know to fo my work.
So yeah, some of us do know that a Gaussian is its own Fourier transform, derivative of an exponential is the exponential itself, etc.
FWIW in college when I would visit family during holidays, I was often confused by cultural references; someone would slip in a catch-phrase and everyone in the room would laugh.
I was too busy to watch TV in those days, and it was like being from another culture.
After college, I had fallen out of the habit of watching TV, and for decades I thought my culture of tech journals and overwork was superior to their sitcoms and crime dramas.
I can still find merit in my choices, but of course that is missing the point. I was wrong. However frivolous the _structure_ of TV entertainment, the _function_ of shared experience reinforced in-group bonds within my family.
I feel like a Martian for having to explain this to myself. So I thank you, dxbydt, for sharing your story. :-)
I love this story! Both mathematics and pop-culture have an abundance of names and terminology that can make their discussion very foreign to people who haven't encountered the lingo.
I've been thinking about this as a way to understand why many people hate math. I think a lot of people tend to miss that learn math is a lot like learning any other lingo. They assume somehow when they don't understand something it's due to their lack of intelligence rather than simple unfamiliarity with the terms being used. "It is well known that..." gets interpreted as "Even a moron would understand that..." rather than "It is a key piece of cultural knowledge of this domain that...".
Now no one would ever make a statement like "It is well known that Katy Perry is a celebrity" as part of discussing American pop-culture. They'd just laugh at you instead for not knowing who she is. Not sure which way is better.
A Jacobian is often used in optimisation. In machine learning, optimisation is often relabelled "learning", because they're finding the minimum of a cost function, also sometimes called an error function.
You may have heard of backpropagation. Backprop is merely a convenient way to compute the Jacobian of a feed-forward neural network. The Jacobian computed by backprop is then used to pick a line search direction to minimise the network's error.
Put it another way, a Jacobian is kind of a multidimensional gradient. It tells you in which multidimensions a function is changing the most rapidly.
no, but most of twitter isn't using knowledge of katy perry's whereabouts either.
Storing garbage info like katy perry kim kardashian etc. in your brain, in my humble opinion, is a tax on your cognition - one pays that tax by being ignorant of elementary methods like fourier transform or multivariable calculus or eigenvalues or trig integrals or T distribution or you get the idea...
I'm quite good at math and found your anecdote funny (yes, I know what a jacobian and a hessian are, and give have lectures on using the fourier transform to professional scientists), but honestly most people, including coders, don't need that, and their lives are not bettered by knowing these things. It is a lot bit pretentious, and not humble at all, to judge people for keeping 'garbage' information.
I have no opinion on what others should deposit in their brains. Am just saying personally if I start keeping track of garbage info like kate and kim, I tend to forget my trig integrals. I have empirically verified that over time. So given a choice, I choose to be blissfully unaware of these cultural cues, so I can store stuff I deem more useful
It is “well known” as in “this was worked out 200 years ago and is a foundational idea for several branches of mathematics and every undergraduate studying science, mathematics, or engineering learns this several times in multiple classes before the end of college, and readers of a mathematician’s blog post about his new research results can be expected to be well familiar with the idea, and if they aren’t the research will not be meaningful to them anyway”.
Please don't post this sort of empty dismissal here. I know it feels funny at the time, but it's boring in the end. There's no insight here, and as jordigh and others pointed out, it's a cheap shot.
Since people sometimes wonder: humor is welcome on HN, lame humor not so much. It chokes out the rarer things, and there are plenty of places to get it elsewhere.
Scroll to the end of his blog entry. There Terry Tao gives an addendum explaining in layman's terms the meaning of their work:
"ADDED LATER: the following analogy (involving functions with just two zeroes, rather than an infinite number of zeroes) may help clarify the relation between this result and the Riemann hypothesis (and in particular why this result does not make the Riemann hypothesis any easier to prove, in fact it confirms the delicate nature of that hypothesis). ..."
Probably should have been placed at the beginning of the blog for the main content is certainly intimidating.
While I thoroughly enjoy Terry Tao's work and am a big admirer of his down-to-earth philosophy, I don't really understand why this particular article is so popular. Indeed, the top comment (which made me LOL, btw) seems to reflect the same attitude.
A much more accesible and enjoyable (though still somewhat technical) article is [this one] [1]
I wonder if it bothers someone like Terry Tao that a question of this requires so much machinery to solve. I'd like to believe him that, say, the work to renormalize the Hamiltonian there is totally correct and rigorous -- he's the best person to believe that it is, anyway. But why is it... necessary?
It seems so completely unlikely that the universe would require going so far afield from the original question in order to prove something that's relatively simple to state about it!
Of course, it's probably because there's some big piece missing -- if you knew how to prove the RH, this would probably be a comparatively trivial corollary. But it still feels kinda weird that all those conclusions couldn't be 'reduced' to something simpler.
Maybe it can be reduced to something simpler, because this is the first proof and proofs are often simplified later, but as far as proofs go this one isn't that complex.
For a more accessible example, what about Fermat's Last Theorem? An apparently simple statement about integer arithmetic, whose proof needs to go into wild excursions about arcane functions of continuous curves. Counterintuitive!
>It seems so completely unlikely that the universe would require going so far afield from the original question in order to prove something that's relatively simple to state about it!
There is nothing in the universe that is simple to prove rigorously.
68 comments
[ 2.9 ms ] story [ 145 ms ] thread"In brief, the Riemann hypothesis is equivalent to the conjecture that Λ ≤ 0."
This result seems to imply that the De Bruijn-Newman constant (Λ) is non-negative, i.e., >=0. Thus, were one able to prove that Λ=0, then one would have also shown the Riemann Conjecture to be true? Similarly, if one were to show Λ to be not equal to 0, then the Riemann Conjecture would be false.
Again, just a quick reading. Sure others who are better/smarter can chime in!
Update: Tao expands on this point in the bottom of his blog post.
Can somebody with more info chime in--- is this a huge, groundbreaking amount of progress?
So really it's hard to say. You might even say this proof made things harder. It guarantees that you have to prove the constant is 0 if you want to prove the conjecture. Before you might have hoped prove negativity.
So, not huge in absolute terms, but on the other hand, huge, as it removes all remaining negative numbers.
You could instead choose a sigma algebra that does not contain {0}. Trivially, you could choose the standard Lebesgue measure restricted to {{}, R}.
A somewhat more useful example would be to identify x with x+1 for all x (or equivalently use the topology on R induced by the map t -> e^2πit from R to S^1), which results in a topology where only periodic functions are continuous.
For a complete measure you get the additional requirement that 0 isn't allowed to be contained in any null-set.
Funny that he should mention it: just the other day, my mom and I were having a good laugh because as it turned out mom's neighbor, a Mrs. Cringleton, apparently had no idea that the Gaussian is its own Fourier transform under these circumstances. Apparently it came up over a cup of tea and bicuits they were having, and Mrs. Cringleton felt quite a bit embarrassed when my mom remarked: "But my dear, that is well-known!"
Poor Mrs. Cringleton...
On the other hand, I have some professors who abused the word "trivial" to mean "it can be proven by me". One professor spent the better half of an hour thinking about a question, only to declare that the answer was trivial. His explanation followed and filled the rest of the hour.
I wonder if the origin of the concept of something being trivial only in hindsight can be dated, or has it always been a truism?
In that case the word 'trivial' means it doesn't require complex machinery to solve. A simple consequence of arithmetic manipulations, say, may be extremely difficult to think of yourself, but once you know it it's trivial to see the validity of. I believe that counts as 'trivial' in mathematical terminology.
One of the things I find fascinating about the Internet is the effect it has on writing when you have so little control over who your audience is.
Every now and then, a blogger or Twitterer used to writing for some niche audience catches the larger eye of the Internet zeitgeist and all of the sudden momentarily finds themselves with a much larger, very different audience, who perceive their work entirely differently because they lack the context the author presumed.
I wonder how much of the modern trends of political correctness, trigger warnings, etc. come from this experience where you can no longer safely make any assumptions about your audience. At the same time, it's not possible to write effectively when you assume the audience could be anyone, even a potentially malicious aggressor.
So, can we recreate the good echo chambers, such as academia, but do it right this time, such that irrelevant characteristics no longer matter but the people inside of them no longer have to deal with nutballs and assholes coming in and disrupting anything?
> tea and bicuits
Where are you claiming to be from?
https://en.wikipedia.org/wiki/Mid-Atlantic_accent
The team I was on was trying to hire a lady for a front end eng role. I didn’t know what the lady’s name was, but I was told she was a very promising candidate, and we were all looking forward to the day when she joined because we had a ton of frontend work.
Ok, so at lunch, my colleague said out aloud - Hey Katy Perry is in the lunchroom! Everyone got excited. So I said, Oh is that the new front end engineer ? The whole room exploded in laughter. I thought I must have made a big social gaffe. My mind quickly raced...,oh maybe she must be a celebrity of some kind. Perry Perry...which Perry do I know...So previous night I was watching some show on TBS and suddenly a Perry name flashed into my brain, so I instantly said - Oh she must be the daughter of Tyler Perry!
The room just doubled up in laughter. Finally I did a wiki lookup and found out who she was. But for days, people would look at me, laugh and say he’s the Katy Perry Tyler Perry guy ha ha ha.
Finally I was so embarrassed and pissed off I asked them - Do you know the difference between between Jacobian and a Hessian ? Nobody knew. Well, I said triumphantly, I might not know your katy perry from tyler perry, but atleast I know what I need to know to fo my work.
So yeah, some of us do know that a Gaussian is its own Fourier transform, derivative of an exponential is the exponential itself, etc.
https://www.xkcd.com/1053/
Aside, I honestly wonder if I would be able to recognize Katy Perry in person...
I was too busy to watch TV in those days, and it was like being from another culture.
After college, I had fallen out of the habit of watching TV, and for decades I thought my culture of tech journals and overwork was superior to their sitcoms and crime dramas.
I can still find merit in my choices, but of course that is missing the point. I was wrong. However frivolous the _structure_ of TV entertainment, the _function_ of shared experience reinforced in-group bonds within my family.
I feel like a Martian for having to explain this to myself. So I thank you, dxbydt, for sharing your story. :-)
I've been thinking about this as a way to understand why many people hate math. I think a lot of people tend to miss that learn math is a lot like learning any other lingo. They assume somehow when they don't understand something it's due to their lack of intelligence rather than simple unfamiliarity with the terms being used. "It is well known that..." gets interpreted as "Even a moron would understand that..." rather than "It is a key piece of cultural knowledge of this domain that...".
Now no one would ever make a statement like "It is well known that Katy Perry is a celebrity" as part of discussing American pop-culture. They'd just laugh at you instead for not knowing who she is. Not sure which way is better.
You may have heard of backpropagation. Backprop is merely a convenient way to compute the Jacobian of a feed-forward neural network. The Jacobian computed by backprop is then used to pick a line search direction to minimise the network's error.
Put it another way, a Jacobian is kind of a multidimensional gradient. It tells you in which multidimensions a function is changing the most rapidly.
Storing garbage info like katy perry kim kardashian etc. in your brain, in my humble opinion, is a tax on your cognition - one pays that tax by being ignorant of elementary methods like fourier transform or multivariable calculus or eigenvalues or trig integrals or T distribution or you get the idea...
What he likely meant is that it is well known from the literature. Or there's an easy explanation that supersedes the verbose notation.
[1] https://www.youtube.com/watch?v=rLDgQg6bq7o
Since people sometimes wonder: humor is welcome on HN, lame humor not so much. It chokes out the rarer things, and there are plenty of places to get it elsewhere.
"ADDED LATER: the following analogy (involving functions with just two zeroes, rather than an infinite number of zeroes) may help clarify the relation between this result and the Riemann hypothesis (and in particular why this result does not make the Riemann hypothesis any easier to prove, in fact it confirms the delicate nature of that hypothesis). ..."
Probably should have been placed at the beginning of the blog for the main content is certainly intimidating.
A much more accesible and enjoyable (though still somewhat technical) article is [this one] [1]
[1] https://www.google.nl/amp/s/terrytao.wordpress.com/2010/04/1...
It seems so completely unlikely that the universe would require going so far afield from the original question in order to prove something that's relatively simple to state about it!
Of course, it's probably because there's some big piece missing -- if you knew how to prove the RH, this would probably be a comparatively trivial corollary. But it still feels kinda weird that all those conclusions couldn't be 'reduced' to something simpler.
There is nothing in the universe that is simple to prove rigorously.