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Math Science

Tracking the World's Great Unsolved Math Mysteries 221

Posted by samzenpus
from the another-piece-of-pi dept.
coondoggie writes "Some math problems are as old as the wind, experts say, and many remain truly unsolved. But a new open source-based site from the American Institute of Mathematics looks to help track work done and solve long-standing and difficult math problems. The Institute, along with the National Science Foundation, has opened the AIM Problem Lists site to offer an organized and annotated collection of unsolved problems, and previously unsolved problems, in a specialized area of mathematics research. The problem list provides a snapshot of the current state of research in a particular research area, letting experts track new developments, and newcomers gain a perspective on the subject."
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Tracking the World's Great Unsolved Math Mysteries

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  • by Oxford_Comma_Lover (1679530) on Wednesday November 18, 2009 @08:13PM (#30151054)
    http://en.wikipedia.org/wiki/Collatz_conjecture [wikipedia.org] Speaking of unsolved math mysteries, the 3n+1 problem is a fabulous way to spend days and days of your life. It's particularly fun if you think about it in binary. Whatever the answer is, it's either simple and elegant or complex beyond imagination.
    • Re: (Score:3, Funny)

      by ae1294 (1547521)

      WHOA... Gotta love that little meme..

      If the starting value n = 27 is chosen, the sequence, listed and graphed below, takes 111 steps, climbing to over 9,000 before descending to 1.

      { 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1 }

      • I noticed that too... LOL.

        That conjecture is interesting and simple enough to be understandable without being a math person. I looked at the wiki article for the above mentioned Riemann Hypothesis, and that's a bit too complex for me.

        • by ae1294 (1547521)

          That conjecture is interesting and simple enough to be understandable without being a math person. I looked at the wiki article for the above mentioned Riemann Hypothesis, and that's a bit too complex for me.

          Yeah same here, not really a math person but you gotta love how simple it would be to write a program to play around with it. Who knows, if you picked the right starting number you might even prove it wrong! but I'm not really sure how one would ever be able to prove it right!

    • by Tynin (634655)
      Wish I hadn't posted in this discussion, I'd love to toss you an interesting mod. This will no doubt steal hours of my life, thanks :-)
    • by Shikaku (1129753)

      This is something Ruby is DESIGNED for.

      http://pastebin.com/m25fc9de4 [pastebin.com]

      I popped this out in a few minutes, but if it can be modified to save every valid Collatz number it finds and not recalculate anything at all it can go pretty fast for very little code and eat all your RAM in the process :)

    • by ACS Solver (1068112) on Wednesday November 18, 2009 @09:51PM (#30151950)
      I have fond memories of that one. On the subject of teaching and education...

      One of my math teachers once showed me the problem. The teacher knew I'm decent at math and would occasionally show me interesting or unusual problems. The interesting part is, the teacher told me to have a try at proving the proposition of this problem, without telling me that it's an unsolved problem. So I had a good amount of fun trying to prove this. Of course, it's not like I could make a proof with my high school knowledge, but it challenged my mind and was a fun thing to do. And had the teacher told me right away that it's an unsolved problem, I wouldn't have had the motivation to think about it, knowing beforehand that I wouldn't be able to find a proof.

      That was one of my educational highlights, though. Way to provide a mental challenge!

      I'm still amazed by how part of the problem's beauty is that it's easy to understand the actual proposition. That isn't true for most unsolved problems, after all. Take the recently proven Poincaré conjecture, just understanding what it states takes some math knowledge, though it has a nice approximation in layman's terms. As for the example of the Hodge conjecture [wikipedia.org], I probably don't know half the mathematical concepts required to understand the problem.
    • Whatever the answer is, it's either simple and elegant or complex beyond imagination.

      Actually, if you believe this guy [arxiv.org], it's not only complex beyond imagination, it's complex beyond any possible finite representation, that is, it's unprovable.

    • Why is the alternative to halving, 3n+1? Why 3? I'm curious. If it were just n+1 it seems like it would converge to 1 pretty quickly (since most non-even numbers become even if you add 1).

      10 gives you:
      10 5 6 3 4 2 1

      100 gives you:
      100 50 25 26 13 14 7 8 4 2 1

      What if it were 4n+1? Then 10 gives you:
      10 5 21 85 341 1365 5461 21845 uh oh

      What if it were 5n+1? Then 10 gives you:
      10 5 26 13 76 38 19 96 48 24 12 6 3 16 8 4 2 1

  • by 140Mandak262Jamuna (970587) on Wednesday November 18, 2009 @08:17PM (#30151080) Journal
    I have this wonderful proof for this conjecture, but unfortunately the 80 char limit for sig in slashdot is too small for it.
  • Sadly... (Score:5, Funny)

    by cosm (1072588) <thecosm3@nospaM.gmail.com> on Wednesday November 18, 2009 @08:25PM (#30151162)
    their servers will explode when they take a stab at Navier-Stokes [wikipedia.org]. I asked Wolfram-Alpha, but it simply returned the exact solution of a degenerate case, the solution being 'Fuck you.'
  • by JoshuaZ (1134087) on Wednesday November 18, 2009 @08:29PM (#30151218) Homepage

    See http://rjlipton.wordpress.com/2009/11/12/more-on-mathematical-diseases/ [wordpress.com] for unsolved problems which are all really simple and also really addicting to think about. For many of these, the best way to stop thinking about one of them is to start thinking about another.

    I'm actually a bit puzzled as to why this is a Slashdot article. If I wanted to point to something new in the way people are doing math I'd point to Math Overflow http://mathoverflow.net/ [mathoverflow.net] where many professional mathematicians, grad students and others are active. It is essentially a centralized system for people to post math questions and get math answers from people who know. It is very cool. It also is highly addictive to read.

  • by Anonymous Coward on Wednesday November 18, 2009 @08:37PM (#30151306)

    As of about a year ago, a new kind of collaborative math project known as "polymath" is emerging. These research projects are completely open for any interested scholar to drop in and make contributions to the problem at hand. The technical infrastructure is based on well-known tools such as wikis and forum discussions

    The very first such project successfully explored a new approach to the density Hales-Jewett thorem--a significant problem in combinatorics--in about six weeks of effort, with a fully preserved record of about a thousand contributions from dozens of participants.

    See Polymath Wiki [michaelnielsen.org] for the details. This new contribution from the AIM will provide a focus point for such efforts and encourage similar massively collaborative projects.

    And of course, the emerging field of computer-verified mathematics [vdash.org] is also dependent on massive collaboration, in order to translate existing results into a fully-formalized form that computes will understand and verify as correct. A wiki-based project could be a great help there as well.

  • by Joe The Dragon (967727) on Wednesday November 18, 2009 @09:17PM (#30151690)

    why not hide them in video games so we can get more people to look at them.

    • by Inda (580031)
      Excellent idea!

      ---

      You enter a dark room. Inside the room is a large door with the words "Entrance to the second level" scratched in the paintwork. Below the door handle is a riddle.

      "Extend the Kronecker-Weber theorem on abelian extensions of the rational numbers to any base number field."
  • by Rockoon (1252108) on Wednesday November 18, 2009 @09:18PM (#30151702)
    Only (very) loosely related but deserving mention is the Encyclopedia of Integer Sequences. [att.com]

    This encyclopedia has proven very useful for me in that I have avoided 'solving' many problems with it.
  • Meh. (Score:4, Insightful)

    by jd (1658) <imipak@yahoo.cEINSTEINom minus physicist> on Wednesday November 18, 2009 @10:10PM (#30152154) Homepage Journal

    Mathematically modelling the brain would seem to be a very trivial problem. The problem is that there's a lot of brain to model. I've posted (admittedly non-rigorous) mathematical models of the brain on Slashdot before, but narry a grant check from it. Bah.

    Computational fluid dynamics for foams, liquid crystals, et al, isn't any harder than for anything else. The equations are chaotic by nature, but chaotic systems can be well-behaved on aggregate under certain conditions. CFD as generally done relies on some specifically hand-picked special case or cases being universally true. They never are, which is why most CFD differs from how systems actually behave in practice.

    If you were to treat CFD as a problem in chaos theory, rather than as isolated collections of imperfect examples of special cases, there would be no problem. It is always when engineers try to take shortcuts and oversimplify the maths to make it easy on themselves that they run into problems. They should be locked up for their own safety. If you want to really annoy them, lock them up with some airgel foam.

    The problem with chaotic systems is that the system is sensitive to initial conditions, which means the only way to get "correct" results is to use infinite precision and zero step sizes. This isn't useful, but is a good way to annoy experts in CFD.

    This leaves two options - use very very big, very very fast computers (the option used by F1 teams), or find an equivalent problem you CAN solve (the idea behind transforms).

    Ok, does chaos look like a good place to use transforms? If you could identify and classify the Strange Attractors in the system, can you do anything useful? Probably not, at least not in the "solving the problem" sense. Chaos is fully deterministic, but it is utterly unpredictable. The only solution is the whole solution.

    What knowing the Strange Attractors might tell you is how to vary the precision and step-size to get the best results for a given level of compute power. But it's going to be all raw horsepower from thereon out.

    The best way to invest money on such work is to design a co-processor that performs a handful of fairly high-level maths functions directly (optimized purely for speed, not physical or logical space) so that you can do Navier-Stokes almost at the level of raw hardware rather than through clunky software. Then cluster the living daylights out of the co-processor.

    It's necessary to optimize commodity hardware for space, because chip real-estate is expensive. However, if you're building what is basically a SOP (single-operation processor) for a dedicated market that can afford things like Earth Simulator, the only time you care about space is when it impacts speed.

    Ideally, if the speed of light wasn't an issue, you'd want each bit in the output to be produced by wholly independent logic, duplicating the input bits as necessary to accomplish this. In practice, you'd probably want to start with that conceptually but in reality have something that was somewhere between that and a highly compressed form. Too parallel and the delays in communication exceed the benefits from the parallelization.

    But this is all obvious. Anyone here who has done multi-threading or any other form of parallelization knows about synchronization issues and communication overheads. It's even one of the biggest chunks of any course on the subject of parallel design. There's nothing new there, certainly nothing "unsolved".

    But, yeah, a well-designed Navier-Stokes co-processor would likely give you greater accuracy and greater performance than the modern pure software solutions. Especially those using ugly protocols to do the communications.

    If Intel can conceptulalize 80 Pentium 4 cores on a wafer, it would seem reasonable enough to imagine modern fabrication methods being able to put at least a couple of hundred dedicated Navier-Stokes processors into the same space. Since the input for an iteration would be based on output from that and other processors, there's no

    • <asshole>

      Mathematically modelling the brain would seem to be a very trivial problem. The problem is that there's a lot of brain to model. I've posted (admittedly non-rigorous) mathematical models of the brain on Slashdot before, but narry a grant check from it. Bah.

      You must be either one of the greatest geniuses of all time, or uninformed on the topic of neurological modeling. People doing real research generally tend not to waste their time trolling Slashdot to find insightful theories, so you might want to try to get it published in a journal instead.

      If you were to treat CFD as a problem in chaos theory, rather than as isolated collections of imperfect examples of special cases, there would be no problem. It is always when engineers try to take shortcuts and oversimplify the maths to make it easy on themselves that they run into problems.

      What specific treatment, pray tell, would suffice as a "one algorithm fits all" approach to solving Navier-Stokes (let alone when mixed with extra behavior like crystal growth, chemical/thermal diffusion

      • by jd (1658)

        People doing useful and interesting research frequently post on Slashdot, so I don't see what your problem is. It doesn't take a genius to mathematically model a brain and that isn't something people have bothered much with doing.

        Some things people have tried to do are build models of compartments of the brain (bad idea), simulations of some poorly-specified upper-level functions of the brain (even worse idea) and discrete/binary simulations of individual neurons assuming them to be stateless and/or with a

    • Re:Meh. (Score:5, Funny)

      by Tomfrh (719891) on Thursday November 19, 2009 @01:52AM (#30153374)

      Mathematically modelling the brain would seem to be a very trivial problem.

      Yours perhaps...

    • If you could identify and classify the Strange Attractors in the system, can you do anything useful? Probably not, at least not in the "solving the problem" sense. Chaos is fully deterministic, but it is utterly unpredictable. The only solution is the whole solution.

      That is not entirely correct.
      First strange attractor [wikipedia.org] is usually embedded into lower-dimension manifold. Reducing dimensionality can make problem a lot more tractable, especially if original system was infinitely-dimensional (like Navier-Sto

      • by jd (1658)

        Ok, I would agree with all that. (I can't mod you informative for obvious and non-chaotic reasons.)

    • by dkf (304284)

      Mathematically modelling the brain would seem to be a very trivial problem. The problem is that there's a lot of brain to model. I've posted (admittedly non-rigorous) mathematical models of the brain on Slashdot before, but narry a grant check from it. Bah.

      You sound like a pure mathematician. (You know what I mean: "a solution has been shown to exist, so it is trivial".)

      The problem is that the brain is a non-linear system on many scales, and it's not clear that the nature of the non-linearity is the same at all scales. This makes even approximate modeling rather difficult. And there's a lot of detail, and a lot of different scales. Right now, it's easier to let poets and psychologists write the higher-level models than to derive them either numerically or ana

  • The problem list provides a snapshot of the current state of research in a particular research area, letting experts track new developments

    So when a promising idea comes along, the "expert" can follow up and hopefully get credit for the solution. I see this is the workplace and on the net in various places. Technical discussion forums are lurked by "experts" in industry who look for ideas without contributing anything to the discussion. Some people don't mind, others don't realize, and others are bothered

  • Hilbert problems (Score:3, Interesting)

    by aws4y (648874) on Wednesday November 18, 2009 @10:43PM (#30152408) Homepage Journal
    I am pretty sure that some of the problems at least will be Hilbert Problems that do not currently have a solution. http://en.wikipedia.org/wiki/Hilbert_problems [wikipedia.org]
  • by Phantasmagoria (1595) <loban.rahman+sla ... m minus caffeine> on Wednesday November 18, 2009 @10:54PM (#30152462)

    This is a great idea. It whould promote more interest in the specific problems and unsolved math problems in general. Besides, more collaboration should result in better research.

  • Here's mine... (Score:2, Interesting)

    by rayharris (1571543)
    P = NP?

    I so want it to be true. Quantum computing is our best hope right now of shedding light on this problem.

    And it's not on their list...

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