Top Mathematician Says He's Solved a 160-Year-Old Maths Problem Worth $1 Million

[BrianHG]

Michael Atiyah, one of the world's most renowned mathematicians showed how he solved the 160-year-old Riemann hypothesis at a lecture on Monday - and he will be awarded US$1 million if his solution is confirmed.

https://www.sciencealert.com/top-mathematician-sir-michael-atiyah-solved-a-160-year-old-1-million-maths-problem-riemann-hypothesis

https://youtu.be/jXugkzFW5qY

https://www.claymath.org/millennium-problems/riemann-hypothesis

[CJay]

If my understanding of the hypothesis is correct then it's going to have a somewhat adverse effect on the security of a lot of encryption products

[Marco]

How many pages is the proof?

[m98]

I call black magic.

[Mattjd]

How many pages is the proof?

1 page

I call black magic.

assuming his proof by contradiction is correct. then it depends, as he said, if you believe proof by contradiction is valid or not.

Proof by contradiction is very powerful, but of course it has its problems.

[Mattjd]

https://math.stackexchange.com/questions/2930742/what-is-the-todds-function-in-atiyahs-paper

[taydin]

it's going to have a somewhat adverse effect on the security of a lot of encryption products

Could be. But when it comes to cryptology, the math is very complex, and the best people involved in that field are mostly working for the government/intelligence agencies. So when that expert says "RSA is an open standard that is peer reviewed", if that "peer" is any good, he's probably also working for the government/intelligence agencies.

Us mere mortals can never really know if a particular encryption algorithm is absolutely secure, unless our math advances to the level of those "peers". And without knowing, we have to assume that all encryption algorithms invented by the "peers" have a back door that can be used by the government/intelligence agencies.

[mtdoc]

Us mere mortals can never really know if a particular encryption algorithm is absolutely secure, unless our math advances to the level of those "peers". And without knowing, we have to assume that all encryption algorithms invented by the "peers" have a back door that can be used by the government/intelligence agencies.

I agree. Though if governments had universal access to all encryption, then they would not need to be pursuing this.

[taydin]

I agree. Though if governments had universal access to all encryption, then they would not need to be pursuing this.

Yes, we can never be sure. All we can do is thought experiments ...

It is possible that a private company implements a new and powerful encryption algorithm independent of the government. In this case the government would want to use the force of law to render that algorithm ineffective for itself.

All the whining of the government about encryption preventing governments from catching bad guys could also be a smoke screen  The governments want people to think that their communication is private, which will yield better intel. I always remember the special browser setting, a checkbox that says "don't track me"  How many people check that box and think they are untracked now? 

[CJay]

it's going to have a somewhat adverse effect on the security of a lot of encryption products

Could be. But when it comes to cryptology, the math is very complex, and the best people involved in that field are mostly working for the government/intelligence agencies.

You'd certainly hope they are.

The way us Brits are supposed to have invented the programmable computer and asymmetric public key encryption but kept it secret  for decades and can't help but wonder if that hypothesis has been cracked a long time ago but been kept under wraps because it's 'useful'.   

[Mattjd]

At the end of the day, even if the proof is valid. It doesnt change much atm. Cryptologists and the like already operate under the assumption that the RH is true. Additionally, the first million primes or so have been computed, following the Riemann zeta function, RH would just give more.

Clever people would have to take the results and try to find ways that to helps exploit current algorithms and what not.

[Cerebus]

If my understanding of the hypothesis is correct then it's going to have a somewhat adverse effect on the security of a lot of encryption products

Not really, one application of the Riemann hypothesis if proved is a way to calculate the number of primes less than a given number, something that at the moment is more a matter of guesswork. It is no help with the innate difficulty of prime factorization of composite numbers, which is the crux of many public key encryption algorithms. It won't break any prime based encryption schemes, but may demonstrate that they are either harder or easier to break than currently believed by putting an upper bound on the difficulty of the prime factorization involved in cracking a particular encryption scheme.

[TerraHertz]

Dear oh dear. Fascinating, but I find his speech cadence nearly impossible to follow, and that tiny 'slides' box is completely unreadable.
Hopefully there'll be a more accessible explanation sometime.

[Cerebus]

Hopefully there'll be a more accessible explanation sometime.

Good luck with that - given that the Riemann hypothesis itself, expressed in its simplest form, is already inaccessible to probably 99% of the population, an accessible explanation of the proof of it is going to be quite a way off.   

A mathematician I know was saying that there are probably 10 or 20 people in the world qualified to validate Atiyah's proof (people who know both their way around the R.H. and the types of mathematics used by Atiyah in his proof).

[BrianHG]

Dear oh dear. Fascinating, but I find his speech cadence nearly impossible to follow, and that tiny 'slides' box is completely unreadable.
Hopefully there'll be a more accessible explanation sometime.

It significantly helps if you both turn on closed captioning and play the video at 1.5x speed.  Not perfect, but, at least you can make it through his un-evenly timed with accent speech.

To see full screen size slides, use this video link: https://hitsmediaweb.h-its.org/Mediasite/Play/35600dda1dec419cb4e99f706197a3951d
And click on "swap video elements" to shrink the speaker and enlarge the slides in the presentation.
These is a bi-directional index seek on this player to each slide.

[Marco]

Proof by contradiction is very powerful, but of course it has its problems.

If math is subtly inconsistent then we have really big problems to begin with.

Since the proof is so short it seems an ideal case for using a proof assistant to me, why rely on a consensus of experts when hard logic suffices?

[Cerebus]

If math is subtly inconsistent then we have really big problems to begin with.

Something Bertrand Russell figured out a long time ago, which led to the reformulation of mathematics on axiomatic lines.

[Vtile]

Math is nothing except subtly inconsistent outside our four basic functions +,-,* & / 

Doesn't make any less useful though

[dmills]

Something Bertrand Russell figured out a long time ago, which led to the reformulation of mathematics on axiomatic lines.

And then along came Godel and broke that (In a very similar way)!

Regards, Dan.

[Marco]

Incompleteness is less of a problem than inconsistent.

[Mattjd]

Math is nothing except subtly inconsistent outside our four basic functions +,-,* & / 

Doesn't make any less useful though

they're operators not functions

[Mattjd]

consistency in math btw is NEVER being able to prove both X and NOT(X) simultaneously

[Tepe]

Math is nothing except subtly inconsistent outside our four basic functions +,-,* & / 

Doesn't make any less useful though

they're operators not functions

That's basically a question of syntax

[T3sl4co1l]

consistency in math btw is NEVER being able to prove both X and NOT(X) simultaneously

Correct.  Which is consequently why Godel proved the eponymous Incompleteness Theorem: either a math system is too simple, i.e. incomplete (and probably not very useful overall); or it's inconsistent (i.e., it is possible to prove true "P AND NOT P").

The argument is often told as a statement in prepositional calculus.  The statement is constructed as an interpreter of prepositional calculus, operating on some given number.  It just so happens, the number is constructed to code for an inconsistent statement.  (A statement in PC can only be built stepwise from the rules of the system; but any finite number can be constructed with finite steps, arbitrarily without the code equivalent being subject to rules.*)  It's the logical equivalent of (and predecessor to) the Halting Problem.

*In modern terms, a sort-of equivalent is coding in any syntactically checked high-level language, versus loading a (relatively large) number that itself just so happens to be machine code.  The code compiled from the language must follow the structure of that language (of course, practical computing languages are not made nearly so limited as mathematical systems are, so it is still possible to express the Halting Problem in, say, a few dozen statements...), whereas the number is completely arbitrary and unstructured (until something happens to read that number as some coded system, and it turns out to have interesting effects).

Tim

[Cerebus]

Math is nothing except subtly inconsistent outside our four basic functions +,-,* & / 

Doesn't make any less useful though

they're operators not functions

That's basically a question of syntax

Mathematically the relationship is actually { x | x is an operator } ⊂ { y | y is a function }, that is, all operators are functions, not all functions are operators. However, operator is used loosely and is very loosely defined (except in the most rigorous textbooks) and in general mathematicians will use operator when the function it represents is being applied to a ring, field or vector space and/or rules of associativity and commutativity are brought into play in respect of functional composition. Basically, if you ask two mathematicians for a dichotomous formal definition of operator and function you will get two different formal answers, but they'd probably agree that an operator is like pornography, they know one when they see one.