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Tracking the World's Great Unsolved Math Mysteries
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."
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
It's a small world and it smells funny; I'd buy another if it wasn't for the money; Take back what I paid (SoM)
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.
Loban Amaan Rahman ==> Anagram of ==> Aha! An Abnormal Man!
I would argue the opposite: a problem is something which has a solution, something without a solution is not a problem but a circumstance.
No, the bicycle is equivalent to a number base or a mathematical system. It is an implementation OF an underlying system (in this case, Newton's Laws), but Newton's Laws would still remain exactly the same whether Newton - or indeed bicycles - had ever existed.
The definition is also immaterial, as that too is an implementation detail. The underlying principle would remain unaltered whether the definitions of circumference, diameter or pi had ever been developed.
You are confusing the overlaid system with what it overlays. I'm saying you don't need to. Your argument is that the overlaid system is artificial, an invented product. I'm saying you're entirely correct on that. But what I am also saying is that what the product overlays, what is beneath the terms, the dynamics and the fancy Greek lettering is not artificial but exists whether it is known to exist or not.
The problem with assuming the two layers are the same is that you run into the Anthropomorphic Principle - the universe is the way it is because it produced people capable of seeing it. Let us, for a moment, assume the Many Worlds theory of Quantum Mechanics is correct. Then there are universes OTHER than the one we see and the theory falls down. The same would be true if the model of a multiverse as a foam (where each universe is a bubble in that foam) is correct.
But if you're on this site, you should be familiar with layering anyway. Maths - the fundamental, overarching thing that is shown in all mathematical systems that exist, will exist or ever have existed - is a Layer 1 concept in the OSI model. Concepts like numbers and other fundamental but artificial building blocks are Layer 2, which makes Group Theory a layer 2 switch. Anything and everything that MUST be true because of something in layer 2 is arguably also layer 2, which would include Goedel's Theorum. Anything that is true only in a specific implementation of mathematics is layer 3 or above.
Does using an OSI representation make it easier to see how not all maths is the same?
It's a small world and it smells funny; I'd buy another if it wasn't for the money; Take back what I paid (SoM)
First of all "as old as the wind" is just an expression means "really fricken old". It's obviously not meant to be taken literally, so get off your high horse.
Secondly,
If it is discovered, the solution already exists and the problem was solved before wind existed
Just because a solution exists, does not mean you have solved the problem. Think of it this way. You are looking through your telescope at night up at the stars and you notice a new star you have never seen before. You look at all the star-charts you can find and realize that no one has ever documented this star. You've just discovered it.
But you are saying that you did not discover the star, since the star already existed. Of course the solutions already exist for these math problems. However, discovery is the act of documenting an observation (ie, someone has to say "this is the answer"), so while they exist, no one has yet discovered them.