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Bored? Need Money? Solve One of the Millennium Prize Problems

02/22/2017 8:43 AM

The 7 Millennium Prize Problems (6 remain unsolved)

In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) established seven Prize Problems. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude.

The prizes were announced at a meeting in Paris, held on May 24, 2000 at the Collège de France. Three lectures were presented: Timothy Gowers spoke onThe Importance of Mathematics; Michael Atiyah and John Tate spoke on the problems themselves.

The seven Millennium Prize Problems were chosen by the founding Scientific Advisory Board of CMI, which conferred with leading experts worldwide. The focus of the board was on important classic questions that have resisted solution for many years.

Following the decision of the Scientific Advisory Board, the Board of Directors of CMI designated a $7 million prize fund for the solutions to these problems, with $1 million allocated to the solution of each problem.

It is of note that one of the seven Millennium Prize Problems, the Riemann hypothesis, formulated in 1859, also appears in the list of twenty-three problems discussed in the address given in Paris by David Hilbert on August 9, 1900.

The problems are...

Yang–Mills and Mass Gap

Experiment and computer simulations suggest the existence of a "mass gap" in the solution to the quantum versions of the Yang-Mills equations. But no proof of this property is known.

Riemann Hypothesis

The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann's 1859 paper, it asserts that all the 'non-obvious' zeros of the zeta function are complex numbers with real part 1/2.

P vs NP Problem

If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem? This is the essence of the P vs NP question. Typical of the NP problems is that of the Hamiltonian Path Problem: given N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that it is correct. But I cannot so easily find a solution.

Navier–Stokes Equation

This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding.

Hodge Conjecture

The answer to this conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations. The Hodge conjecture is known in certain special cases, e.g., when the solution set has dimension less than four. But in dimension four it is unknown.

Poincaré Conjecture (Solved by Grigori Yakovlevich Perelman)

In 1904 the French mathematician Henri Poincaré asked if the three dimensional sphere is characterized as the unique simply connected three manifold. This question, the Poincaré conjecture, was a special case of Thurston's geometrization conjecture. Perelman's proof tells us that every three manifold is built from a set of standard pieces, each with one of eight well-understood geometries.

Birch and Swinnerton-Dyer Conjecture

Supported by much experimental evidence, this conjecture relates the number of points on an elliptic curve mod p to the rank of the group of rational points. Elliptic curves, defined by cubic equations in two variables, are fundamental mathematical objects that arise in many areas: Wiles' proof of the Fermat Conjecture, factorization of numbers into primes, and cryptography, to name three.

The list of the problems can be found here.

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#1

Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/22/2017 9:42 AM

This is very interesting... and I hate to distract and apologize for it, so I'll mark it off-topic.

This reminds me of a father and son team (Erik Demaine and Martin Demaine, both teach at MIT) that solved a Origami problem where one could not the paper into. They did solve it, with one cut, as I recall, but not through. I first saw these two about 15-20 years ago. About the time Erik was entering MIT.

Erik is now a professor at MIT (Computer Science) and applies Origami ( computational origami ) to a lot of principles, such microbiology in cell construction.

I just watched a documentary of him, and he was finishing off a paper that basically is a formula for constructing folds to paper for any object.

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#2
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Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/22/2017 9:45 AM

<...a formula for constructing folds to paper for any object...> Hasn't 3-D printing put an end to all that?

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#3
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Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/22/2017 9:52 AM

No, origami is to 3D printers as is you need to crawl before you can walk.

as your get into Origami and understanding the actually science, it goes beyond that. for applications. Such as packing solar arrays for deployment on rocket launches to more down to earth as transportable protective barriers for law enforcement.

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#4
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Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/22/2017 9:58 AM

That is really interesting. I found his webpage here:

http://erikdemaine.org/

If you can remember the name of the documentary, let me know, I'd like to watch it. Cool Stuff!

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#5
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Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/22/2017 10:51 AM

I think it was called "Between the Folds" and Netflix is where I saw it.

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#6
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Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/22/2017 10:59 AM

I think I found a link to it here. There is an ad in the begining and an intro but then I think it launches into the documentary. I'll have to watch it when I have time. Looks cool.

http://documentaryheaven.com/between-folds-art-of-origami/

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#7
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Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/22/2017 12:22 PM

This reminds me of a father and son team (Erik Demaine and Martin Demaine, both teach at MIT) that solved a Origami problem where one could not the paper into.

Confusing sentence. Did you mean 'cut the paper in two'?

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#8
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Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/22/2017 12:55 PM

Yes, it was. I was going off of memory (not a good thing). When I first saw it about 15 years ago.... at first, I thought it was just a one cut 'not through'

but as it turned out, you were allowed 'one cut', after it was folded, all the way through..., after reviewing the link bayes supplied, it made sense.

I should have done some research before posting. Unfortunately, when I see something interesting,... I tend to jump right in.

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#9

Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/22/2017 2:46 PM

I'm curious.... the Poincaré Conjecture (Solved by Grigori Yakovlevich Perelman)

Society Prize (1991)
EMS Prize (1996), declined
Fields Medal (2006), declined
Millennium Prize (2010), declined (Poincaré Conjecture)

What I'm curious about, is why he declined these awards?

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#11
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Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/22/2017 3:42 PM

According to Perelman, his reasons for turning down these prizes are: "the main reason is my disagreement with the organized mathematical community. I don't like their decisions, I consider them unjust."

From what I've read, he doesn't think all mathematicians are bad, just that some have ethical issues (the awarding of prizes can be very political). He resents the fact that most mathematicians are willing to go along to get along with the system. He's an idealist who does mathematics out of love and is disappointed others aren't as passionate and pure in their love of mathematics.

https://en.wikipedia.org/wiki/Grigori_Perelman#The_Fields_Medal_and_Millennium_Prize

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#12
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Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/22/2017 5:21 PM

"From what I've read, he doesn't think all mathematicians are bad, just that some have ethical issues (the awarding of prizes can be very political). He resents the fact that most mathematicians are willing to go along to get along with the system. He's an idealist who does mathematics out of love and is disappointed others aren't as passionate and pure in their love of mathematics."

Rather reminds me of my two runs through the higher education system. Of both experiences and having had over a dozen assorted math classes covered by more than that many professors and student teachers combined I have to say that as of now two out of the whole bunch stand out in my mind as having been good decent people.

The other dozen plus in my opinion ranged from being simple and out of touch with reality to outrightly dangerously unethical and or highly emotionally and mentally unstable and it was very obvious the system that employed them did not care given how many complaints some of them had against them that were common knowledge amongst other faculty and students.

Hell, one of my math professors who prided himself of being so good with students had his own brother stab a kid to death and the professor himself when questioned over it by authorities scored so badly on the investigation questioning he got pulled in and ran though a psych eval only to be then given an indefinite leave of absence from the college for being way to out of touch with reality (and self control) to be trusted in a classroom to say the least.

Personally from about 15 minutes into the first class with him I had a very bad feeling he was bonkers and probably worse. And from there I can say I had at least 5 more over my college experiences I felt just as uneasy about after one or two classes with them as well.

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#13
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Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/22/2017 8:51 PM

'....

it was very obvious the system that employed them did not care given how many complaints some of them had against them that were common knowledge amongst other faculty and students.

Hell, one of my math professors who prided himself of being so good with students had his own brother stab a kid to death and the professor himself when questioned over it by authorities scored so badly on the investigation questioning he got pulled in and ran though a psych eval only to be then given an indefinite leave of absence from the college for being way to out of touch with reality (and self control) to be trusted in a classroom to say the least

....'

.

Immediately after stating 'the system..' '...did not care', you present strong evidence to the contrary.

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#14
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Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/23/2017 6:09 AM

That is your claim solely. Where is your proof to make this claim? Provide a suitable link to back up your statement of contrary strong evidence.

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#17
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Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/23/2017 9:29 PM

No link needed. I copied and pasted the evidence.

.

Come on, you have to be able to see the difference between your claims of the UK outlawing diesels with NO backing to be found in or out of the links you provided, and my effort to point out that in the narrative the school actually did take action to protect students and staff, despite claims they did not care.

Don't get mad when I point out inconsistencies/possible errors. Just make an effort to address the problems.

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#10

Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/22/2017 3:18 PM

Shouldn't encryptions like 'one-time-pad' disprove 'P vs NP' conjecture as a problem that is difficult to solve, but easy to check.

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#15

Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/23/2017 2:06 PM

Some years back I provided the art history research for a chemical engineering seminar on "The Art of Pissarro and the Navier-Stokes Equation." I didn't know that N-S hadn't been proved though. I figured it was old news.

The connection between Camille Pissarro and Navier-Stokes: he was the first artist, at least in the ken of the guy who did the actual ChemE part of the talk, who represented smokestack plumes accurately -- even though Pissarro is considered an Impressionist. These were the days when the Industrial Revolution was spreading rapidly. Before then no one painted smokestack plumes because they didn't exist. See "The River Oise near Pointoise" (1873, painting in the Clark Institute collection).

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#16

Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/23/2017 4:53 PM

I looked at these 7 questions and I am puzzled ( no pun intended )

I looked on the internet for some answers, but could not find any.

Can someone here offer an explanation that us laypersons could understand.

Here are my two questions:

1. Did the mathematicians ( not sure if that is the correct word ) that came up with these " riddles" work on them their whole lives and finally went to the grave without ever solving them ?

2. If these " riddles " were solved, in what particular area would the information be applied toward ( offer a tangible example (s).

Can an answer be provided starting with the first, as posted, concluding with the last, in order.

There are a great many things offered up for discussion on cr4 and those that are, I find to be the most interesting.

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#18
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Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/23/2017 9:45 PM

1. Yang Mills/Mass gap: in a simplified (perhaps over) version this problem arises from the inability of currently accepted models to reconcile the strong predictive utility of Yang Mills with the disallowed (current understanding) combination of particles with mass traveling at the speed of light.

This problem relates toour fundamental understanding physics. The applications could be very widespread depending on what is ultimately discovered. New types of sensors? New materials? New ways to communicate? New forms of propulsion or power generation? Difficult to say.

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#19
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Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/23/2017 9:57 PM

2 Reinmann hypothesis: this is solidly in the realm of number theory. It deals with the distribution of prime numbers. While the hypothesis has been verified to be correct out to huge numbers, no solid proof hs been devised.

I am not sure what immediate practical applications would gain from such a proof, but as this is one of the foundations of number theory, a proof in this area might unfold numerous proofs in other areas of number theory. Cryptography might be affected early on.

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#20
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Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/23/2017 10:03 PM

3. P vs NP problem. The click able link hs a good description of this problem. Even so, I believe my understanding of this problem is probably flawed, as I don't yet understand why certain encryption techniques aren't a valid demonstration of the existence of problems with are not easily solved but for which solutions are easily checked.

Applications would likely include encryption.

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#21
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Re: Bored? Need Money? Solve One of the Millennium Prize Problems

02/23/2017 10:17 PM

4. Navier-Stokes deals with fluid flow. The Navier-Stokes existence and smoothness problem is the lack of a proof or disproof that the Navier-Stokes equation has a smooth solution everywhere.

Additional knowledge gleaned about Navier-Stokes could have wide applications from more efficient cars boat and planes, to progress in plasma physics and fusion reactors, even to changes in cardiovascular operations.

.

Perhaps someone else could take over from here....

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