Yay for unelected politicians who keep coming back despite being forced to resign in shame.
I think his unelected status is a bit of a red herring, to be honest. The unelected Lord Mandelson of Hartlepool and Foy is pretty much exactly as dodgy as the duly elected Rt Hon Peter Mandelson MP was. Plus there are plenty of other very dodgy elected politicians (as the recent expenses furore demonstrated), and lots of entirely sensible and honest unelected peers (especially the non-party-affiliated cross-benchers) who do very good work, are acknowledged experts in their fields, and who are definitely worth having involved in the legislature. But yes, there does seem to have been a bit of a tradition of "two strikes and you get a peerage" in British politics over the last few decades: the novelist and convicted liar Jeffrey Archer is a prime example, and it can only be a matter of time before David Blunkett gets kicked upstairs too.
You can't knot ('nontrivially embed') 1-dimensional string in 4-space, because the fourth dimension (degree of freedom) means you can get rid of the knot by lifting a small section of the string perpendicular to the other three dimensions, which means there's now a gap in the string in the original 3-dimensional 'slice' of 4-space, through which you can pass any awkward strands of the knot.
But, although you can't knot 1-dimensional string in 4-space, you can knot 2-dimensional surfaces. This is less easy to visualise, but one way to get started is to picture a knotted length of string sweeping out a surface. A good place to start if you're interested in visualising higher dimensions is 'The Fourth Dimension' by the SF author, logician and mathematician Rudy Rucker.
I work part-time for a couple of mathematics research journals and we do get the occasional crank submission. There's one guy who's been sending us, on average, a 'paper' every week or so for the past few years: typically a single, badly-written page of gibberish (we're talking Time Cube standard lunacy here) which is clearly not the work of someone who's ever seen a real mathematics paper. We've never responded to him, or even acknowledged any of his submissions (helpfully he prints his return address on the back of the envelope, so these days they go straight in the bin, unopened and unread) and yet he still keeps sending them in.
The arXiv also tends to get its fair share of crank submissions, usually elementary attempted (but trivially broken) proofs of things like the Goldbach Conjecture, Fermat's Last Theorem and the like - I'm assuming that the really mad stuff is filtered out by the moderators.
In contrast, at a quick glance to my nonspecialist eyes (I'm a knot theorist) Xian-Jin Li's preprint looks like a genuine (if flawed) attempt by a serious, qualified mathematician who specialises in the relevant area. Fair play to him for trying, though. I'm also not sure I'd characterise Terence Tao or Alain Connes' refutations as 'rebukes' - they looked more like dispassionate analyses of the paper's flaws to me, the sort of discussion you'd expect from the peer-refereeing process.
Remember kids, it just THEORY. This slogan is currently being used to disastrous (and in a lot of cases, utterly and reprehensibly dishonest) effect by the `Intelligent Design' lot, who have been very successfully playing on the subtle but important difference between the colloquial and scientific meanings of the word `theory'.
Colloquially, to say that something's a theory means that it's an idea somebody's had, a suggested but unproved model of how (some small aspect of) the world works.
In a scientific context, however, a theory is a carefully tested model which has thus far passed every test that anybody's thrown at it. The way it works is that you come up with a hypothesis about how something might work, and then you (and everyone else working in the field) do your absolute level best to knock it to pieces, by doing any and all experiments you can thing of which might highlight errors or flaws in the model. If one of these experiments does break your model then you modify it to take account of the results, or (if you can't do that) you start again from square one with a different model.
Einstein's special and general theories of relativity have been quite extensively tested over the past century, to the point where we're confident that they describe a macroscopic model of the universe that is consistent with every experimental observation obtainable with current technology.
It was known for some time that Newton's theories did not completely accurately describe the universe - careful observation showed that Mercury's orbit precesses at a slightly different rate to that predicted by the Newtonian model. The discrepancy is pretty small - about 40 seconds of arc per century - but it's the principle of the thing. Einstein later said that possibly the finest moment of his scientific career was when he plugged the numbers into his model, and found that they predicted this extra discrepancy absolutely spot on.
It's entirely possible that there are (macroscopic) flaws in Einstein's theories - and if/when they're discovered, the race will be on to refine the model further (just as Einstein's theory was a refinement of Newton's theory).
Saying it's `only a theory' belies the decades of very careful testing and research that has gone into checking and rechecking the model.
I don't understand the finer details myself, but here goes. The 1- and 2-dimensional cases are relatively easy to prove, but the case n=3 is sufficiently complicated that those methods aren't sophisticated enough.
Similarly, although the higher-dimensional (n>4) cases are progressively more complicated still (with weirder and weirder behaviour possible), more sophisticated techniques become feasible, which don't work in lower dimensions. Also, a lot of interesting questions in 3- or 4-dimensions become trivial in higher dimensions.
For example: You can knot a (1-dimensional, embedded) string in 3-dimensional space - and indeed there's a rich branch of mathematics which is devoted to studying the different ways you can do this. But you can't knot a string in 4-dimensional space, because there're are sufficient degrees of freedom to allow you to untie it.
This is tricky to visualise, but think of the 3-dimensional `slice' of 4-space which the knot is resting in. Take a small section of the string and push it slightly in the fourth direction. In the main 3-dimensional slice, there's now a gap in the string, which we can use to untie the knot - pass the knotted bits of the string through this gap until we're left with a straight length of string with a gap in it. Now push the `missing' bit of string back down into the main 3-dimensional subspace. At no point have we cut or glued the string, but we've untied the knot.
(There's an excellent and very readable book on higher dimensions: The Fourth Dimension (And How To Get There) by Rudy Rucker.)
Basically, what's happening here is that there's enough `room' (for want of a better word) in 4-space to make the interestingly nontrivial behaviour we see in 3-space (namely the possibility for knotting 1-dimensional strings) fail to happen.
And, from what little I understand of the details of the Poincaré Conjecture, there's an analogous thing happening there - the techniques which enable the solution of the higher-dimensional versions of the Conjecture aren't applicable in lower-dimensional space (specifically 3-space) because (in some hand-wavy, technically imprecise sense, at least) they rely on having a little bit more room to manoeuvre.
This is why 3- and 4-dimensional topology is quite a popular field - lots of interesting nontrivial things happen which don't happen in higher dimensions.
This sounds like a typical excuse for universities to get more grant (much of which is tax) money - delusions of practical application "if only we get just a little more money, we could do so much!" (the cries of which are repeated every year).
This is a valid concern, and you're certainly not the first person to voice it - indeed, it seems to have become not just ok, but actively encouraged for government education ministers to rant about how taxpayers' money is being wasted on frivolous matters (before getting into their chauffeur-driven Jaguars and heading off to sumptuous gourmet lunches).
It's something I've thought a lot about, myself - I've spent thousands and thousands of pounds of my own, hard-earned and carefully-saved money over the past eight years, funding myself through an MSc and then a PhD in pure mathematics. Not because I thought it would boost my salary, or anything, but because I wanted to learn more about something which interested me. I don't begrudge a penny or second of the money or time I spent doing it - it changed my life, it changed the way I think about everything, and I learned a lot of fascinating stuff along the way.
What it boils down to, I guess, is that if you want to live in a cultured, advanced society (and you should, if only because of the fringe benefits like the general population being healthier and longer-lived, and the crime rate being lower, and so on - but also because it makes life a lot more interesting) then you have to pay people to find out new stuff. Some of this will be practical things like engineering or applied science, some of it will be less practical stuff (like pure science or mathematical research) which might at some point turn out to have practical applications (and both mathematics and `blue-skies' scientific research has a pretty good record in this regard), and some of it will be entirely impractical stuff like art or music.
We simply can't rely on private corporations to fund this sort of stuff - with a few notable exceptions (such as Xerox PARC) they generally have an intensely short-term, practical viewpoint on research - if it's not going to start bringing in hard cash within a couple of years then they don't tend to bother. And there's no particular reason why they should - that's not what they're for.
So what does that leave? Charities and central government, basically. And central government means our tax money.
Do I begrudge my tax money being spent on dubious private finance deals where some private company profits immensely from knocking down a perfectly serviceable hospital, selling off the land, and building a new, smaller, less well-equipped hospital further away? Well yes, actually, I do. Quite a lot, in fact.
Am I irked that a phenomenal amount of UK taxpayers' money is being used to wage a dubious (and catastrophic) war halfway round the world, for the benefit of Texas oil-barons and the Halliburton board of directors? Very much so.
Do I begrudge the Arts Council using (a far tinier fraction of) my tax money to fund Damien Hirst to pickle a sheep's carcass in a tank of formaldehyde? Not for a second. I might think he's a nutter for doing it, and I might not necessarily want to see it when he's done it, but I'm glad to live in a civilisation where people are trying out things like that, for no other reason than they think it might look nice, or interesting, or encourage people to think or look at the world in a slightly different way.
Does a Mozart string quartet directly generate money? No - or at least not much, and not for very many people. But it enhances the world in so many ways, and inspires and motivates the people who listen to it. There are numerous anecdotal cases of important scientific insights being triggered by a piece of music or a painting.
Mathematics is a bit of a hybrid, really. It's `blue skies' research, done purely for its own sake, but it has an unreasonably good track record of turning out to be really useful someti
I'll try and explain what the Conjecture is, because it's not entirely obvious. First of all, I need to explain what the 3-sphere is.
The n-sphere (which mathematicians generally denote by S^n) can be thought of as `all points in (n+1)-dimensional space which are at unit distance from the origin'. So S^2 is the surface of a solid 3-dimensional ball. This sometimes surprises people, who expect this to be S^3 but the key observation here is that the 2 refers to the intrinsic dimension of the object, rather than the extrinsic dimension of any space you might happen to put (`embed') the object in. The fact that we often think of the 2-sphere as being embedded in 3-dimensional space doesn't change the fact that it's inherently a 2-dimensional object. An ant wandering around on it still only has two degrees of freedom.
The 3-sphere (S^3) locally looks like ordinary, flat, Euclidean 3-space, but on a larger scale it kind of doubles back on itself - if you keep walking (or floating) in a `straight line' (well, actually the 3-dimensional analogue of a `great circle', but never mind) in any direction, then you'll eventually get back to where you started.
The Poincaré Conjecture says
Any homotopy 3-sphere is homeomorphic to the 3-sphere
This, by itself, isn't particularly enlightening to the non-topologist, but what it actually boils down to is:
Any closed, compact, simply-connected 3-manifold is homeomorphic to the 3-sphere
What does this mean?
Well, an `n-manifold' is a space which locally looks like ordinary, flat, Euclidean n-dimensional space. So a 3-manifold is a space (like S^3) which locally looks like ordinary 3-space (but which might twist back on itself in a peculiar way on a larger scale). `Closed' means that the 3-manifold doesn't have a boundary - no matter how far you walk, you're not going to run into a brick wall, or fall off the end. `Compact' is a bit more technical, but in this context essentially means you don't get odd shooting-off-to-infinity stuff you have to deal with.
And `simply-connected' means that the first homotopy group (the `fundamental group' of the space) is trivial. What that means is that any closed loop (of string, if you like), in the manifold, can be continuously shrunk down to a point. Here `continuous' means that you're not allowed to cut or glue the string while you're doing it.
To use a 2-dimensional analogy, the 2-sphere (the surface of the 3-dimensional ball, remember, or alternatively a British doughnut) is simply-connected, because given any closed loop in the surface, you can shrink it down to a point without it getting snagged on anything. Whereas the 2-torus (the surface of an American doughnut) isn't, because you can't shrink all closed loops down to a point - one which goes all the way round the central hole, for example, can't be shrunk.
Finally, `homeomorphic' is basically a technical word for `topologically equivalent' - we allow continuous deformations (stretching, twisting, etc, but not cutting or pasting), rotations, reflections, or any combination of these.
So, the (classical) Poincaré Conjecture is essentially a technical way of saying ``If it looks like a 3-sphere then, basically, it is''. (For certain definitions of `is', and `looks like'.)
The analogous conjecture in n-dimensional space is known to be true for n=1 (trivial), 2 (pretty simple), and 5 and above (the 5-dimensional case was proved by Zeeman, who is my PhD grandsupervisor - my supervisor was one of his students). The 4-dimensional case is weird, and there are three different forms to consider - the `piecewise linear' and `topological' cases have been proved, but the `smooth' case is still unproven.
As I understand it, what Perelman claims to have done is prove Thurston's Geometrisation Conjecture, which implies the Poincaré Conjecture as a special case - rather lik
No, a British billion is 10^12, too (although the American definition seems to have largely supplanted the British one in popular usage). We have a (now obscure) word `milliard' which means 10^9.
Presumably this means that Bill Gates is actually a `milliardaire' when visiting the UK.
For a nation which often claims to be the greatest country in the world, I must say that American billions are a bit, well, small...
Britain is the World's Surveillance Leader Yeah. Yeah, that sounds about right.
I guess the idea about CCTV cameras is that in principle the benefits (in terms of greater public safety) in theory outweigh the disadvantages (in terms of infringed privacy).
But for this to be a valid argument (and I'm really not sure I'm convinced by it just yet), there are a number of things which have to happen.
Firstly, the things have to be very strictly controlled and regulated, preferably by some suitably impartial but trustworthy non-governmental organisation. I don't want to see private companies running the things, either. And any abuse of the system must be punishable by serious sanctions - not just a £100 fine and a slap on the wrist, but something like a 2-10 year prison sentence for the individual, and summary revocation of the operating organisation's licence.
And secondly, they have to be shown to actually fulfil the purpose they're set up for. Now a number of people have described incidents where they've totally failed to work. So at present, I'm tending towards the view of them not being a Good Thing - but if these issues can be resolved properly then I'd be (very cautiously) ok about them.
Actually, there was a case a year or two ago where a guy got severely depressed (due to a number of things that had recently happened in his life) and was standing on the edge of a bridge, contemplating jumping off. Somehow, a nearby CCTV camera spotted him, and the police were dispatched to talk him down. So far, all well and good - tragic suicide averted, friends and relatives spared immense grief and feelings of guilt, etc. And then the footage was plastered all over the national news, but without the customary pixelisation and identity-concealment. So suddenly tens of millions of people knew all about a particularly unfortunate and private episode in his life. Which can't have helped matters much, certainly.
Now given our current Home Secretary's attitude towards privacy, I'm not convinced that proper regulation and oversight is going to be high on the list of priorities. David Blunkett (who is currently pushing really hard for the introduction of a compulsory biometric national id card and accompanying national population database) is pretty much the most authoritarian, control-freak Home Secretary we've had since Michael Howard (who held the post under during the last Tory administration, and is now Leader of the Opposition).
A postgrad chemistry student friend of mine told me of something that happened to a colleague of his (another postgrad in the same department). He'd turned up to the department, nice and early, ready for a solid day's work on whatever it is chemistry postgrads do all day. Locked his bicycle up in the rack in the departmental car-park, in full view of the prep room.
Came back at about 6pm, to find bike missing. Wanders around for a bit, double-checking that he hadn't locked it up somewhere else and forgotten. Gives up and knocks on window of prep room. Technician wanders over and opens window.
``Hi - what's the matter?'' ``My bike's missing - you didn't happen to see anyone dodgy hanging around out here today, did you?'' ``No... sorry.'' replies the technician. ``Wait, there was that guy at about 2...'' interjects his colleague, who's just wandered over. ``Oh. No... surely... Erm... Oh...'' replies the first technician. ``What guy?'' asks postgrad cyclist. ``Well, there was one guy earlier on, who was having a lot of difficulty with his bike lock - keys were stuck or something. So we... erm...'' ``Yes?'' enquires cyclist. ``We... lent him some tools...''
Sometimes even sensible precautions are no match for a confident thief who looks as though he should be doing what he's doing.
Actually, I believe Nation's original description of the Daleks (in his original 1963 script) was pretty vague. The job of turning his brief notes into reality was given to Raymond Cusick, of the BBC design department. I think I read somewhere that he was paid a flat fee of forty pounds for designing the Daleks - Terry Nation, however, owned the copyright and did very nicely out of it.
Exactly who in their right mind would buy SCO Unix?
Well I've been wondering this for several years, actually. Last time I used it, it fell over abruptly and spectacularly, with the kernel dumping core in/var/log/syslog.
Perhaps I'm a bit old-fashioned about these things, but I don't consider that a reasonable way for an OS to behave.
We took to calling it `Skoda Unix' around the office - which I now realise was deeply unfair to that much-maligned brand of car.
If we can prove the Poincare Conjecture is true OK, yes, that would definitely be worth doing. Various chunks of 3-dimensional geometric topology currently include the caveat ``... if the 3-dimensional Poincare Conjecture is true.''
(and it is now thought that it might be) Depends who you talk to, of course. Perelman reckons he's got an outline proof of Thurston's Geometrisation Conjecture, which implies the Poincare Conjecture as a corollary. But as I understand it he's not actually proved it, just described how one would go about proving it - essentially saying ``It's over the other side of that hill''. Now Perelman is a very bright lad (cleverer than I am) but the Poincare Conjecture has a long history of almost being proved (my research supervisor (I've just finished a PhD in algebraic and geometric topology) almost did it about twenty years ago, for example, but there was a very subtle and unresolvable flaw in his argument) so I'm going to reserve judgement until the paper turns up in the Annals of Mathematics.
I wouldn't be surprised if the Conjecture turned out to be true, as the corresponding result is known to be true in dimensions 1,2,4,5,... (well, to be picky, the smooth 4-dimensional case hasn't been proved yet, but the topological and piecewise-linear ones have). But equally, I wouldn't be shocked if it weren't - 3-dimensional topology is a weird subject, as there's enough room to start discussing interesting things, but there's not so much room that everything trivialises.
Quick precis of the Conjecture: ``If it looks like a sphere, it is''.
Slightly less quick precis of the Conjecture: Given any topological space X, we can assign a sequence of groups (`homotopy groups') pi_n(X) to it, so that pi_n(X) in some sense describes the n-dimensional structure of X. An n-manifold (a topological space which locally `looks like' ordinary, flat n-dimensional space) which has the same sequence of homotopy groups as the 3-dimensional sphere, is called a `homotopy 3-sphere'. The Poincare Conjecture is that the 3-sphere itself is the only homotopy 3-sphere.
So, to prove it, we have to show that every homotopy 3-sphere is topologically equivalent (`homeomorphic') to the 3-sphere. And to disprove it, we just have to find one counterexample - a homotopy 3-sphere which isn't equivalent to the 3-sphere itself.
Now to use a computer to prove the Conjecture, we need to find some way of verifying that every possible homotopy 3-sphere is equivalent to the 3-sphere. This is theoretically doable - there's an algorithm (the Rego-Rourke algorithm) which lists (with redundancy), all possible homotopy 3-spheres. There's another algorithm (the Rubenstein-Thompson algorithm) which, given a homotopy 3-sphere, can tell if it is equivalent to the 3-sphere. So in theory we just feed the output of the RRA into the RTA.
Except that there are an infinite number of possible homotopy 3-spheres to check, so if the Conjecture is true, this program will never terminate.
Now if you can find some way of reducing the cases under consideration to some finite subset (by, for example, showing that all but a finite number of homotopy 3-spheres obviously satisfy the conjecture) then using a computer suddenly becomes a worthwhile endeavour. This is basically what Appel and Haken did with the Four-Colour Map Theorem.
The computer approach is also useful for searching for counterexamples, and for verifying that all cases up to some level of complexity satisfy the Conjecture.
But where computers aren't currently going to help, is in the actual creative side of things - a lot of important modern mathematics (and the related theorems and proofs) has come from very clever humans saying ``OK, what happens if we try this (utterly weird and counterintuitive) thing here'' and having the aesthetic sense to tell when something's actually interesting or just irrelevant. Until computers are capable of this kind of creative/intuitive/aesthetic/etc behaviour, I doubt they're going to be replacing human research mathematicians any time soon.
I feel I should mention that Isaac Newton (or even Issac Newton, whoever the hell he was) wasn't a PhD - such things didn't exist in the UK educational system in the 17th century. According to the MacTutor History of Mathematics Archive he got his BA in 1665 and incepted MA in 1668, at which point he was elected a college Fellow.
There have been higher doctorates (which, unlike in the US, are not always honorary awards) in the UK for centuries - doctorates in Divinity, Law, Medicine, and Music date back to mediaeval times, while the DSc and DLitt came in in the late 19th century. But the PhD, being a degree somewhere between the first postgraduate degree (studied at the beginning of one's specialist academic career) and the higher doctorates (typically awarded late in a career, on the basis of a substantial published research record), is less than a century old in the UK.
The PhD originated in Germany sometime in the 16th century, migrated to the US sometime in the 19th century (I think) and was introduced in the UK (to some initial scepticism) in the early 20th century.
These days, it's pretty much impossible to get anywhere in (British) academia without a PhD, but that's only really been the case in the last thirty years. In Newton's day (and this seems to have been true at Oxford and Cambridge until the mid-20th century) things were less rigidly qualification-focused. Being elected a Fellow of a college was the important thing.
A few years later, he was appointed Lucasian Professor of Mathematics - a post held currently by Stephen Hawking, and at various other times by Airy, Babbage, Stokes and Dirac.
More importantly, though, Newton was a Fellow of the Royal Society which pretty much beats any other academic honour short of a Nobel Prize or a Fields Medal.
One common criticism of the use of vellum is that of animal cruelty, although it's worth noting that the goatskin used is a by-product of the leather industry, and comes from goats that had already been slaughtered to make, say, shoes.
I was glad to hear that this latest attempt at pointless `modernisation' (for the sake of appearing `modern' rather than for any deeper and more sensible reasons) was defeated. Not least because it's really cool to be able to say ``This is actually an Act of Parliament from the 16th century, and that's actually Henry VIII's signature - not a photocopy, not a JPEG, but the real thing.''
A related matter concerns the increasing prevalence of digital photography. As this BBC News article explains, digital photography may cause problems for future generations of local or family historians. Proper (printed) photographs tend to get stored away in shoeboxes in attics, and are still more-or-less as legible after a hundred years as they were when they were taken. Whereas an entire collection of digital photographs can be wiped out by one hard-disk failure.
Or maybe in fifty years' time nobody'll know how to display a JPEG. Somewhere I've got a tape with the very first program I wrote, recorded on it. It was a simple bullseye game for the Video Genie II (a TRS80 clone) - it wasn't particularly sophisticated by general standards, but not at all bad for an eight-year-old kid. I have no means of retrieving it on any of the small menagerie of computers currently in my house.
Another, related, example: in the mid-1980s a (for the time, pretty damned impressive) multimedia project was launched - the Domesday Project. This was a laserdisc containing digital reproductions of the original 11th-century Domesday Book itself (a census survey of the entirety of England ordered by William I) together with (I think) the 1981 national census data. All very innovative (albeit rather a costly system) but ironically, 15 years on, the laserdiscs are not readable by current technology - but the original 900-odd-year-old Domesday Book itself still is.
I guess the point is that it's all very well saying ``this media is guaranteed to last for fifty years'' (although personally I'd be happier with something that'll last several hundred) but also you have to guarantee that the data format itself is going to remain readily decodeable. This is not a problem for 1000-year-old documents on vellum (as long as your Latin or Norman French is ok, and you've done a basic course in paleography).
If mathematicans aren't really interested in helping understand the world, why should society fund them?
This is an important question, and in my opinion has two particularly valid answers.
The first of these is the one that usually gets advanced - that (as with other pure scientific disciplines) we just don't know what `useless' knowledge might turn out to be useful or vital in fifty year's time. This is all well and good, and a perfectly decent reason to study something.
The other one, which I've come to believe more strongly over the past few years, is that which is often advanced in support of arts funding - that it benefits a society greatly (often in intangible and undefinable ways) to study and research things whether or not they have any practical use.
This is a point which, in the UK at least, a succession of education ministers have either missed or fundamentally disagreed with over the past few decades.
Last month, Charles Clarke, the current Secretary of State for Education made some very disturbing comments about how he didn't see the point in spending taxpayers' money on maintaining a group of ``mediaeval seekers after truth''.
He was initially misquoted as saying he didn't see the point in the study of mediaeval history, which rightly got a lot of historians angry, but a later statement clarified that he actually didn't see the point in studying any subject which didn't have a direct positive contribution to UK industrial or economic interests. Which I find even more disturbing - it's understandable (even ok) for the Chancellor of the Exchequer to have such a viewpoint, but I like to think that the Secretary for Education should at least see some worth in all of the education system.
A friend of mine (an eminent evolutionary and reproductive biologist who's also helped design aliens for people like Anne McCaffrey, Larry Niven and Jerry Pournelle, and co-written a couple of books with Terry Pratchett) once said
``Most people think that the end-product of a PhD is a neatly-typeset hardback thesis. It's not - the end product of the PhD is the person who's done the PhD''
which I rather agree with. Studying or researching any subject changes the way you look at the world - often for the better. It teaches you new or variant modes of thought which you can then apply (often unconsciously) to other areas of interest.
For example: A former office-mate of mine now works for the NHS Breast Cancer Screening Service. The topic of her thesis (permutation group theory) is irrelevant to what she does now. But I find it tremendously reassuring to know that there are people that well-educated, and who have been trained to such a high level in thinking clearly and carefully, involved in something that important and worthwhile.
I've also heard `MSRI' pronounced `emissary', which is perhaps a nicer way of looking at things - a bunch of people bringing news and information to a wider world.
On the other hand, though, there is a peculiarly masochistic element to research - especially in something as abstract as pure mathematics. It can be tremendously frustrating and misery-inducing at times, but when it's going ok it's great - in a strangely aesthetic sort of way. From what I've read of other artists talking about their work, it's not unlike writing, painting, sculpture, or composing in this regard.
Personally, I think it's really cool when a bit of mathematics I'm interested in turns out to have some applications in other areas of knowledge, or even in the `real world' (whatever that means). But that's not why I study the subject - I study it because I find it interesting for its own sake.
I barely finished H.S. and I couldn't afford college. I learnt Calculus, Number Theory, Analytic Geometry, and Topology on my own. No college, no professors, no help, just an HP-48, pencil, and paper.
Wow. That takes some considerable motivation and commitment, and is an achievement worthy of respect. Congratulations!
I was worried when I originally wrote that post that it might sound a bit arrogant - I can only reiterate that it wasn't meant like that. This is, of course, a problem with text-based discussion forums - the finer nuances of the conversation can easily become lost. Sorry about that.
I should point out that I don't think that advanced mathematics is beyond an intelligent and motivated person - you yourself are a prime example of that (did I mention how impressive I think that is?). And I think that the majority of topics can be explained to the interested nonspecialist (New Scientist and Scientific American successfully do this all the time, for example).
But deeper and deeper levels of understanding require one to know about more and more subtle concepts, which requires more investment in terms of study. Some cutting-edge research is reasonably easy to explain to non-specialists. But some isn't - not because the researchers don't understand it well enough themselves, and not because the interested non-specialists aren't intelligent enough. It's because some of the concepts involved are extremely subtle and technical, and hence take some sort of advanced study (whether within a formal academic environment, or under ones own steam) before one can really get ones head around what it all means.
This, I suspect, is why there have been a couple of Slashdot discussions about the Riemann Hypothesis (because most computery types have some idea what a complex number is), or newspaper articles about the Fermat-Wiles Theorem or the Goldbach Conjecture (because both of those can be easily stated in terms that non-mathematicians can understand - everyone's been taught Pythagoras at some point, and everyone knows what a prime number is).
On the other hand, I've not seen any newspaper articles about the Hodge Conjecture, Yang-Mills Theory, or the Poincare Conjecture. Because it's difficult to explain to the general public, in a paragraph, what a `projective algebraic variety' or a `homotopy 3-sphere' is. And much less why they should care, or why some weird nutters get excited about them.
And that's a shame. Because those of us who do study mathematics to postgraduate level know it's a fascinating subject.
Personally, I'm in favour of mandatory photo licenses for driving in the UK. It seems entirely reasonable that the old-style photo-less documents be phased out in favour of the new type. The old-style documents do date back a fair way, and society (especially the proportion of the population driving cars) and technology has moved on a little.
I am, however, quite opposed to the introduction of general ID cards for UK citizens, and wrote to the Home Office last week to outline my reasons for this.
The important distinction, you see, is that driving (despite what the majority of motorists seem to think) is not a right - it's a privilege (which should, in my opinion, be withdrawn from about 50% of current UK motorists).
I have various forms of photo-id of my own - a university library card (which I'm required to present whenever I want to borrow books from the university library), a railcard (which I'm required to present in order to claim a 33% student discount for UK rail travel), a passport (which enables me to travel to other countries ), an international student card (which secures me various extra privileges and discounts in participating shops and countries), and a university film society lifetime membership card (which lets me and a guest watch any film for free).
I don't, however, see why I should suddenly (or, more likely, gradually) be required to have an identity card in order to simply reside and go about my normal daily business, in the country that I (and at least several preceding generations of my family) was born in.
If you want to try something of Eddings, beware. There are two 5-piece series and two 3-piece series.
Strictly speaking, there's just one series. He just keeps republishing it with different titles, and different names for some of the characters, that's all.
So right now I'm sitting in a computer lab (working on an overdue assignment). There is a large sign posted (where everyone can see it) that says "No Cellphones!".
The problem is that most people mentally add the amendment ``... except for you'' to signs like this.
Thus, ``No parking'' becomes ``No parking except for you'', for example.
Virgin Trains, in the UK, started designating one carriage as a `Quiet Coach' in which mobile 'phones and personal stereos were prohibited.
Nobody takes the slightest notice.
I've stopped travelling in the quiet coach now, because I found that mobile 'phones ringing there irritate and distract me far more than they do if I'm in one of the normal coaches.
countries like the EU states, China or Russia. Countries which definitely don't care about international problems and the mind-stunning threat of internation terrorism.
Without wishing to get into a political argument, as a citizen of a `second world' country, I'm a bit irked by your implication that nobody outside the USA cares about international terrorism. No disrespect intended to the thousands who died and were bereaved by the horrific attack on the WTC, but the rest of the world has been living with the effects of terrorism for a bit longer than the last year.
To take an illustrative example, over the last thirty years in the UK we've had actual elected Members of Parliament assassinated by both loyalist and republican Irish activists. Shopping centres have been bombed at Christmas, pubs blown up, ordinary people shot for associating with other ordinary people who happen to be `on the other side'.
I count myself fortunate that I've not been personally affected by such things, and I have every sympathy for those who have - like millions of people around the world I watched in horror as the WTC collapsed.
But to claim that countries and federations like the EU, China and Russia (which between them have roughly 1.5 thousand million inhabitants - a *quarter* of the world's population) just aren't interested in international terrorism is at once naive, insular, and offensive, and in the long run will only serve to fuel the sort of ill-feeling and fanaticism which causes the terrorism in the first place.
The world will be a mess and it will be because of whiners like you destroying the worldwide US surpremancy.
The world is a mess partly *because of* the USA's powerful influence. Don't get me wrong, there's a lot about the USA which I admire, and I don't think the UK has a particularly shining track record as far as international diplomacy goes, either, but simply dismissing everything as the other guy's problem is not the right way to go about making the world (or even the USA) a better place.
But then I'm a bleeding-heart liberal eco-head living in a `second world EU state', so what would I know?
I'm glad someone mentioned G&T, and not just because my PhD supervisor is one of the managing editors, and my MSc supervisor is the other one:)
If I remember correctly, the whole thing was sparked off by Rob Kirby's article on the pricing of research journals. There's an interesting article by Joan Birman in the Notices of the AMS (vol 4, no. 7, Aug 2000, pp770-774) which discusses the various issues, and includes detailed discussion of the day-to-day overheads of running a free, properly peer-refereed research journal. It's available from her web page, in PostScript form.
G&T (and its sister journal Algebraic and Geometric Topology, and the related Monograph series) isn't some low-quality vanity-press thing - it's a real, proper, peer-refereed journal with high standards. At a quick glance, I recognise the names of three Fields medallists on the editorial board, as well as some other very eminent names in the field. And yet it's being run with virtually no overheads by two university lecturers (one of whom is semi-retired) in addition to their normal departmental duties (lecturing, administration, supervising research students).
I understand that a lot of the procedure is automated, with a mixture of TeX and Perl, with copies of all articles being submitted to the arXIv.
Ah yes, I'd almost forgotten about the arXiv. A central repository for research preprints in mathematics, physics, and computer science. It's an unrefereed archive for research announcements, preliminary reports, and preprints. Papers submitted to refereed journals often take up to a couple of years to actually appear in print, so the idea is that you issue a preliminary version of your paper to faster communicate your ideas to anyone else who might be interested.
This stuff is great - it's all about collaborative research and the free and efficient sharing of ideas, and it gives me a great sense of hope for the future.
Could you elaborate and tie this in with the number of primes between m and n?
I'm a little less confident about this, but here goes...
As I understand it (and bear in mind that I've not done any complex analysis for several years, and number theory has never really been my forte) sometime during the 19th century Gauss noticed that the distribution of primes was approximated pretty well by a function he called the `logarithmic integral'.
Li(x) is defined as the integral from 0 to x of (1/t) dt. And apparently the number of primes below x (usually denoted pi(x)) is pretty well approximated by Li(x).
Now this is where I start to lose track of things. As I understand it, it was proved at the beginning of the 20th century that the validity of the Riemann hypothesis is equivalent to the assertion that the deviation of Li(x) from the actual value of pi(x) is of the order of sqrt(x)*ln(x).
If I remember correctly, some work of the three great British mathematicians Hardy, Littlewood, and Hardy-Littlewood showed that pi(x) actually oscillates around Li(x) infinitely many times (although it really doesn't do it very quickly - the first value of x for which the graphs cross is very big indeed).
As to whether there's a relatively simple proof out there, I don't know. My (non-specialist) suspicion is that there isn't, because some really clever people have tried to find one for a century and a half and failed. I'd be interested and impressed to be proved wrong on this, though.
Andrew Wiles' celebrated proof of Fermat's Last Theorem (if you haven't done so, read Simon Singh's book on the subject, and if possible watch the BBC Horizon documentary - the transcript is available) was pretty complicated and, I'm told (algebraic geometry isn't my field either - there's an awful lot of diverse mathematics out there) introduced some genuinely new ideas and methods.
The general feeling is that if there were a simple proof of either Fermat or Riemann (or, for that matter, the Goldbach or Poincare Conjectures) then someone would have found it by now - some really top brains have worked on all of these over the years (including several Fields medallists, FRSes, and the like).
(There's a guy who posts regularly to sci.math who reckons he's got a simple proof of Fermat which doesn't resort to all that scary stuff about elliptic curves or modular forms. The consensus seems to be that he's a nutter, though - his `proofs' contain obvious flaws which he refuses to acknowledge, claiming instead the existence of an enormous academic conspiracy against his work.)
It's also often the case (and this was true for the Fermat theorem) that proofs of such intractible problems, even those which are subtly flawed, introduce new ideas and methods of attack.
This is why otherwise sensible mathematicians have a go at these problems - even if they don't manage to solve them, the chances are that the attempt will inspire them to find new methods or potentially important partial results. Even had Wiles' original (flawed) proof turned out to be irrepairable, it was a pretty major piece of work which introduced some important new ideas which could well be useful in solving related problems.
My guess (as an interested non-specialist) is that while a proof of RH would be complicated and elegant, it would also involve some new twist or idea. As for who might do it, my money would be on Prof Louis de Branges of Purdue University - he demolished the (similarly intractible) Bieberbach Conjecture in the 1980s and thus seems to know what he's doing. Or it might be someone else entirely, someone who's spent seven years locked in their attic (as Wiles did).
Slashdot Mirror
Yay for unelected politicians who keep coming back despite being forced to resign in shame.
I think his unelected status is a bit of a red herring, to be honest. The unelected Lord Mandelson of Hartlepool and Foy is pretty much exactly as dodgy as the duly elected Rt Hon Peter Mandelson MP was. Plus there are plenty of other very dodgy elected politicians (as the recent expenses furore demonstrated), and lots of entirely sensible and honest unelected peers (especially the non-party-affiliated cross-benchers) who do very good work, are acknowledged experts in their fields, and who are definitely worth having involved in the legislature. But yes, there does seem to have been a bit of a tradition of "two strikes and you get a peerage" in British politics over the last few decades: the novelist and convicted liar Jeffrey Archer is a prime example, and it can only be a matter of time before David Blunkett gets kicked upstairs too.
Strings snag on 4 corners of 24 hour time cube. Time cube knot will destroy evil humanity.
I'm so sorry, I don't know what came over me. I do apologise.
You can't knot ('nontrivially embed') 1-dimensional string in 4-space, because the fourth dimension (degree of freedom) means you can get rid of the knot by lifting a small section of the string perpendicular to the other three dimensions, which means there's now a gap in the string in the original 3-dimensional 'slice' of 4-space, through which you can pass any awkward strands of the knot.
But, although you can't knot 1-dimensional string in 4-space, you can knot 2-dimensional surfaces. This is less easy to visualise, but one way to get started is to picture a knotted length of string sweeping out a surface. A good place to start if you're interested in visualising higher dimensions is 'The Fourth Dimension' by the SF author, logician and mathematician Rudy Rucker.
I work part-time for a couple of mathematics research journals and we do get the occasional crank submission. There's one guy who's been sending us, on average, a 'paper' every week or so for the past few years: typically a single, badly-written page of gibberish (we're talking Time Cube standard lunacy here) which is clearly not the work of someone who's ever seen a real mathematics paper. We've never responded to him, or even acknowledged any of his submissions (helpfully he prints his return address on the back of the envelope, so these days they go straight in the bin, unopened and unread) and yet he still keeps sending them in.
The arXiv also tends to get its fair share of crank submissions, usually elementary attempted (but trivially broken) proofs of things like the Goldbach Conjecture, Fermat's Last Theorem and the like - I'm assuming that the really mad stuff is filtered out by the moderators.
In contrast, at a quick glance to my nonspecialist eyes (I'm a knot theorist) Xian-Jin Li's preprint looks like a genuine (if flawed) attempt by a serious, qualified mathematician who specialises in the relevant area. Fair play to him for trying, though. I'm also not sure I'd characterise Terence Tao or Alain Connes' refutations as 'rebukes' - they looked more like dispassionate analyses of the paper's flaws to me, the sort of discussion you'd expect from the peer-refereeing process.
Remember kids, it just THEORY.
This slogan is currently being used to disastrous (and in a lot of cases, utterly and reprehensibly dishonest) effect by the `Intelligent Design' lot, who have been very successfully playing on the subtle but important difference between the colloquial and scientific meanings of the word `theory'.
Colloquially, to say that something's a theory means that it's an idea somebody's had, a suggested but unproved model of how (some small aspect of) the world works.
In a scientific context, however, a theory is a carefully tested model which has thus far passed every test that anybody's thrown at it. The way it works is that you come up with a hypothesis about how something might work, and then you (and everyone else working in the field) do your absolute level best to knock it to pieces, by doing any and all experiments you can thing of which might highlight errors or flaws in the model. If one of these experiments does break your model then you modify it to take account of the results, or (if you can't do that) you start again from square one with a different model.
Einstein's special and general theories of relativity have been quite extensively tested over the past century, to the point where we're confident that they describe a macroscopic model of the universe that is consistent with every experimental observation obtainable with current technology.
It was known for some time that Newton's theories did not completely accurately describe the universe - careful observation showed that Mercury's orbit precesses at a slightly different rate to that predicted by the Newtonian model. The discrepancy is pretty small - about 40 seconds of arc per century - but it's the principle of the thing. Einstein later said that possibly the finest moment of his scientific career was when he plugged the numbers into his model, and found that they predicted this extra discrepancy absolutely spot on.
It's entirely possible that there are (macroscopic) flaws in Einstein's theories - and if/when they're discovered, the race will be on to refine the model further (just as Einstein's theory was a refinement of Newton's theory).
Saying it's `only a theory' belies the decades of very careful testing and research that has gone into checking and rechecking the model.
I don't understand the finer details myself, but here goes. The 1- and 2-dimensional cases are relatively easy to prove, but the case n=3 is sufficiently complicated that those methods aren't sophisticated enough.
Similarly, although the higher-dimensional (n>4) cases are progressively more complicated still (with weirder and weirder behaviour possible), more sophisticated techniques become feasible, which don't work in lower dimensions. Also, a lot of interesting questions in 3- or 4-dimensions become trivial in higher dimensions.
For example: You can knot a (1-dimensional, embedded) string in 3-dimensional space - and indeed there's a rich branch of mathematics which is devoted to studying the different ways you can do this. But you can't knot a string in 4-dimensional space, because there're are sufficient degrees of freedom to allow you to untie it.
This is tricky to visualise, but think of the 3-dimensional `slice' of 4-space which the knot is resting in. Take a small section of the string and push it slightly in the fourth direction. In the main 3-dimensional slice, there's now a gap in the string, which we can use to untie the knot - pass the knotted bits of the string through this gap until we're left with a straight length of string with a gap in it. Now push the `missing' bit of string back down into the main 3-dimensional subspace. At no point have we cut or glued the string, but we've untied the knot.
(There's an excellent and very readable book on higher dimensions: The Fourth Dimension (And How To Get There) by Rudy Rucker.)
Basically, what's happening here is that there's enough `room' (for want of a better word) in 4-space to make the interestingly nontrivial behaviour we see in 3-space (namely the possibility for knotting 1-dimensional strings) fail to happen.
And, from what little I understand of the details of the Poincaré Conjecture, there's an analogous thing happening there - the techniques which enable the solution of the higher-dimensional versions of the Conjecture aren't applicable in lower-dimensional space (specifically 3-space) because (in some hand-wavy, technically imprecise sense, at least) they rely on having a little bit more room to manoeuvre.
This is why 3- and 4-dimensional topology is quite a popular field - lots of interesting nontrivial things happen which don't happen in higher dimensions.
This sounds like a typical excuse for universities to get more grant (much of which is tax) money - delusions of practical application "if only we get just a little more money, we could do so much!" (the cries of which are repeated every year).
This is a valid concern, and you're certainly not the first person to voice it - indeed, it seems to have become not just ok, but actively encouraged for government education ministers to rant about how taxpayers' money is being wasted on frivolous matters (before getting into their chauffeur-driven Jaguars and heading off to sumptuous gourmet lunches).
It's something I've thought a lot about, myself - I've spent thousands and thousands of pounds of my own, hard-earned and carefully-saved money over the past eight years, funding myself through an MSc and then a PhD in pure mathematics. Not because I thought it would boost my salary, or anything, but because I wanted to learn more about something which interested me. I don't begrudge a penny or second of the money or time I spent doing it - it changed my life, it changed the way I think about everything, and I learned a lot of fascinating stuff along the way.
What it boils down to, I guess, is that if you want to live in a cultured, advanced society (and you should, if only because of the fringe benefits like the general population being healthier and longer-lived, and the crime rate being lower, and so on - but also because it makes life a lot more interesting) then you have to pay people to find out new stuff. Some of this will be practical things like engineering or applied science, some of it will be less practical stuff (like pure science or mathematical research) which might at some point turn out to have practical applications (and both mathematics and `blue-skies' scientific research has a pretty good record in this regard), and some of it will be entirely impractical stuff like art or music.
We simply can't rely on private corporations to fund this sort of stuff - with a few notable exceptions (such as Xerox PARC) they generally have an intensely short-term, practical viewpoint on research - if it's not going to start bringing in hard cash within a couple of years then they don't tend to bother. And there's no particular reason why they should - that's not what they're for.
So what does that leave? Charities and central government, basically. And central government means our tax money.
Do I begrudge my tax money being spent on dubious private finance deals where some private company profits immensely from knocking down a perfectly serviceable hospital, selling off the land, and building a new, smaller, less well-equipped hospital further away? Well yes, actually, I do. Quite a lot, in fact.
Am I irked that a phenomenal amount of UK taxpayers' money is being used to wage a dubious (and catastrophic) war halfway round the world, for the benefit of Texas oil-barons and the Halliburton board of directors? Very much so.
Do I begrudge the Arts Council using (a far tinier fraction of) my tax money to fund Damien Hirst to pickle a sheep's carcass in a tank of formaldehyde? Not for a second. I might think he's a nutter for doing it, and I might not necessarily want to see it when he's done it, but I'm glad to live in a civilisation where people are trying out things like that, for no other reason than they think it might look nice, or interesting, or encourage people to think or look at the world in a slightly different way.
Does a Mozart string quartet directly generate money? No - or at least not much, and not for very many people. But it enhances the world in so many ways, and inspires and motivates the people who listen to it. There are numerous anecdotal cases of important scientific insights being triggered by a piece of music or a painting.
Mathematics is a bit of a hybrid, really. It's `blue skies' research, done purely for its own sake, but it has an unreasonably good track record of turning out to be really useful someti
The n-sphere (which mathematicians generally denote by S^n) can be thought of as `all points in (n+1)-dimensional space which are at unit distance from the origin'. So S^2 is the surface of a solid 3-dimensional ball. This sometimes surprises people, who expect this to be S^3 but the key observation here is that the 2 refers to the intrinsic dimension of the object, rather than the extrinsic dimension of any space you might happen to put (`embed') the object in. The fact that we often think of the 2-sphere as being embedded in 3-dimensional space doesn't change the fact that it's inherently a 2-dimensional object. An ant wandering around on it still only has two degrees of freedom.
The 3-sphere (S^3) locally looks like ordinary, flat, Euclidean 3-space, but on a larger scale it kind of doubles back on itself - if you keep walking (or floating) in a `straight line' (well, actually the 3-dimensional analogue of a `great circle', but never mind) in any direction, then you'll eventually get back to where you started.
The Poincaré Conjecture says
This, by itself, isn't particularly enlightening to the non-topologist, but what it actually boils down to is:
What does this mean?
Well, an `n-manifold' is a space which locally looks like ordinary, flat, Euclidean n-dimensional space. So a 3-manifold is a space (like S^3) which locally looks like ordinary 3-space (but which might twist back on itself in a peculiar way on a larger scale).
`Closed' means that the 3-manifold doesn't have a boundary - no matter how far you walk, you're not going to run into a brick wall, or fall off the end. `Compact' is a bit more technical, but in this context essentially means you don't get odd shooting-off-to-infinity stuff you have to deal with.
And `simply-connected' means that the first homotopy group (the `fundamental group' of the space) is trivial. What that means is that any closed loop (of string, if you like), in the manifold, can be continuously shrunk down to a point. Here `continuous' means that you're not allowed to cut or glue the string while you're doing it.
To use a 2-dimensional analogy, the 2-sphere (the surface of the 3-dimensional ball, remember, or alternatively a British doughnut) is simply-connected, because given any closed loop in the surface, you can shrink it down to a point without it getting snagged on anything. Whereas the 2-torus (the surface of an American doughnut) isn't, because you can't shrink all closed loops down to a point - one which goes all the way round the central hole, for example, can't be shrunk.
Finally, `homeomorphic' is basically a technical word for `topologically equivalent' - we allow continuous deformations (stretching, twisting, etc, but not cutting or pasting), rotations, reflections, or any combination of these.
So, the (classical) Poincaré Conjecture is essentially a technical way of saying ``If it looks like a 3-sphere then, basically, it is''. (For certain definitions of `is', and `looks like'.)
The analogous conjecture in n-dimensional space is known to be true for n=1 (trivial), 2 (pretty simple), and 5 and above (the 5-dimensional case was proved by Zeeman, who is my PhD grandsupervisor - my supervisor was one of his students). The 4-dimensional case is weird, and there are three different forms to consider - the `piecewise linear' and `topological' cases have been proved, but the `smooth' case is still unproven.
As I understand it, what Perelman claims to have done is prove Thurston's Geometrisation Conjecture, which implies the Poincaré Conjecture as a special case - rather lik
No, a British billion is 10^12, too (although the American definition seems to have largely supplanted the British one in popular usage). We have a (now obscure) word `milliard' which means 10^9.
Presumably this means that Bill Gates is actually a `milliardaire' when visiting the UK.
For a nation which often claims to be the greatest country in the world, I must say that American billions are a bit, well, small...
Britain is the World's Surveillance Leader
Yeah. Yeah, that sounds about right.
I guess the idea about CCTV cameras is that in principle the benefits (in terms of greater public safety) in theory outweigh the disadvantages (in terms of infringed privacy).
But for this to be a valid argument (and I'm really not sure I'm convinced by it just yet), there are a number of things which have to happen.
Firstly, the things have to be very strictly controlled and regulated, preferably by some suitably impartial but trustworthy non-governmental organisation. I don't want to see private companies running the things, either. And any abuse of the system must be punishable by serious sanctions - not just a £100 fine and a slap on the wrist, but something like a 2-10 year prison sentence for the individual, and summary revocation of the operating organisation's licence.
And secondly, they have to be shown to actually fulfil the purpose they're set up for. Now a number of people have described incidents where they've totally failed to work. So at present, I'm tending towards the view of them not being a Good Thing - but if these issues can be resolved properly then I'd be (very cautiously) ok about them.
Actually, there was a case a year or two ago where a guy got severely depressed (due to a number of things that had recently happened in his life) and was standing on the edge of a bridge, contemplating jumping off. Somehow, a nearby CCTV camera spotted him, and the police were dispatched to talk him down. So far, all well and good - tragic suicide averted, friends and relatives spared immense grief and feelings of guilt, etc. And then the footage was plastered all over the national news, but without the customary pixelisation and identity-concealment. So suddenly tens of millions of people knew all about a particularly unfortunate and private episode in his life. Which can't have helped matters much, certainly.
Now given our current Home Secretary's attitude towards privacy, I'm not convinced that proper regulation and oversight is going to be high on the list of priorities. David Blunkett (who is currently pushing really hard for the introduction of a compulsory biometric national id card and accompanying national population database) is pretty much the most authoritarian, control-freak Home Secretary we've had since Michael Howard (who held the post under during the last Tory administration, and is now Leader of the Opposition).
A postgrad chemistry student friend of mine told me of something that happened to a colleague of his (another postgrad in the same department). He'd turned up to the department, nice and early, ready for a solid day's work on whatever it is chemistry postgrads do all day. Locked his bicycle up in the rack in the departmental car-park, in full view of the prep room.
Came back at about 6pm, to find bike missing. Wanders around for a bit, double-checking that he hadn't locked it up somewhere else and forgotten. Gives up and knocks on window of prep room. Technician wanders over and opens window.
``Hi - what's the matter?''
``My bike's missing - you didn't happen to see anyone dodgy hanging around out here today, did you?''
``No... sorry.'' replies the technician.
``Wait, there was that guy at about 2...'' interjects his colleague, who's just wandered over.
``Oh. No... surely... Erm... Oh...'' replies the first technician.
``What guy?'' asks postgrad cyclist.
``Well, there was one guy earlier on, who was having a lot of difficulty with his bike lock - keys were stuck or something. So we... erm...''
``Yes?'' enquires cyclist.
``We... lent him some tools...''
Sometimes even sensible precautions are no match for a confident thief who looks as though he should be doing what he's doing.
Actually, I believe Nation's original description of the Daleks (in his original 1963 script) was pretty vague. The job of turning his brief notes into reality was given to Raymond Cusick, of the BBC design department. I think I read somewhere that he was paid a flat fee of forty pounds for designing the Daleks - Terry Nation, however, owned the copyright and did very nicely out of it.
Exactly who in their right mind would buy SCO Unix?
/var/log/syslog.
Well I've been wondering this for several years, actually. Last time I used it, it fell over abruptly and spectacularly, with the kernel dumping core in
Perhaps I'm a bit old-fashioned about these things, but I don't consider that a reasonable way for an OS to behave.
We took to calling it `Skoda Unix' around the office - which I now realise was deeply unfair to that much-maligned brand of car.
If we can prove the Poincare Conjecture is true
OK, yes, that would definitely be worth doing. Various chunks of 3-dimensional geometric topology currently include the caveat ``... if the 3-dimensional Poincare Conjecture is true.''
(and it is now thought that it might be)
Depends who you talk to, of course. Perelman reckons he's got an outline proof of Thurston's Geometrisation Conjecture, which implies the Poincare Conjecture as a corollary. But as I understand it he's not actually proved it, just described how one would go about proving it - essentially saying ``It's over the other side of that hill''. Now Perelman is a very bright lad (cleverer than I am) but the Poincare Conjecture has a long history of almost being proved (my research supervisor (I've just finished a PhD in algebraic and geometric topology) almost did it about twenty years ago, for example, but there was a very subtle and unresolvable flaw in his argument) so I'm going to reserve judgement until the paper turns up in the Annals of Mathematics.
I wouldn't be surprised if the Conjecture turned out to be true, as the corresponding result is known to be true in dimensions 1,2,4,5,... (well, to be picky, the smooth 4-dimensional case hasn't been proved yet, but the topological and piecewise-linear ones have). But equally, I wouldn't be shocked if it weren't - 3-dimensional topology is a weird subject, as there's enough room to start discussing interesting things, but there's not so much room that everything trivialises.
Quick precis of the Conjecture:
``If it looks like a sphere, it is''.
Slightly less quick precis of the Conjecture:
Given any topological space X, we can assign a sequence of groups (`homotopy groups') pi_n(X) to it, so that pi_n(X) in some sense describes the n-dimensional structure of X. An n-manifold (a topological space which locally `looks like' ordinary, flat n-dimensional space) which has the same sequence of homotopy groups as the 3-dimensional sphere, is called a `homotopy 3-sphere'. The Poincare Conjecture is that the 3-sphere itself is the only homotopy 3-sphere.
So, to prove it, we have to show that every homotopy 3-sphere is topologically equivalent (`homeomorphic') to the 3-sphere. And to disprove it, we just have to find one counterexample - a homotopy 3-sphere which isn't equivalent to the 3-sphere itself.
Now to use a computer to prove the Conjecture, we need to find some way of verifying that every possible homotopy 3-sphere is equivalent to the 3-sphere. This is theoretically doable - there's an algorithm (the Rego-Rourke algorithm) which lists (with redundancy), all possible homotopy 3-spheres. There's another algorithm (the Rubenstein-Thompson algorithm) which, given a homotopy 3-sphere, can tell if it is equivalent to the 3-sphere. So in theory we just feed the output of the RRA into the RTA.
Except that there are an infinite number of possible homotopy 3-spheres to check, so if the Conjecture is true, this program will never terminate.
Now if you can find some way of reducing the cases under consideration to some finite subset (by, for example, showing that all but a finite number of homotopy 3-spheres obviously satisfy the conjecture) then using a computer suddenly becomes a worthwhile endeavour. This is basically what Appel and Haken did with the Four-Colour Map Theorem.
The computer approach is also useful for searching for counterexamples, and for verifying that all cases up to some level of complexity satisfy the Conjecture.
But where computers aren't currently going to help, is in the actual creative side of things - a lot of important modern mathematics (and the related theorems and proofs) has come from very clever humans saying ``OK, what happens if we try this (utterly weird and counterintuitive) thing here'' and having the aesthetic sense to tell when something's actually interesting or just irrelevant. Until computers are capable of this kind of creative/intuitive/aesthetic/etc behaviour, I doubt they're going to be replacing human research mathematicians any time soon.
nicholas
I feel I should mention that Isaac Newton (or even Issac Newton, whoever the hell he was) wasn't a PhD - such things didn't exist in the UK educational system in the 17th century. According to the MacTutor History of Mathematics Archive he got his BA in 1665 and incepted MA in 1668, at which point he was elected a college Fellow.
There have been higher doctorates (which, unlike in the US, are not always honorary awards) in the UK for centuries - doctorates in Divinity, Law, Medicine, and Music date back to mediaeval times, while the DSc and DLitt came in in the late 19th century. But the PhD, being a degree somewhere between the first postgraduate degree (studied at the beginning of one's specialist academic career) and the higher doctorates (typically awarded late in a career, on the basis of a substantial published research record), is less than a century old in the UK.
The PhD originated in Germany sometime in the 16th century, migrated to the US sometime in the 19th century (I think) and was introduced in the UK (to some initial scepticism) in the early 20th century.
These days, it's pretty much impossible to get anywhere in (British) academia without a PhD, but that's only really been the case in the last thirty years. In Newton's day (and this seems to have been true at Oxford and Cambridge until the mid-20th century) things were less rigidly qualification-focused. Being elected a Fellow of a college was the important thing.
A few years later, he was appointed Lucasian Professor of Mathematics - a post held currently by Stephen Hawking, and at various other times by Airy, Babbage, Stokes and Dirac.
More importantly, though, Newton was a Fellow of the Royal Society which pretty much beats any other academic honour short of a Nobel Prize or a Fields Medal.
nicholas
One common criticism of the use of vellum is that of animal cruelty, although it's worth noting that the goatskin used is a by-product of the leather industry, and comes from goats that had already been slaughtered to make, say, shoes.
I was glad to hear that this latest attempt at pointless `modernisation' (for the sake of appearing `modern' rather than for any deeper and more sensible reasons) was defeated. Not least because it's really cool to be able to say ``This is actually an Act of Parliament from the 16th century, and that's actually Henry VIII's signature - not a photocopy, not a JPEG, but the real thing.''
A related matter concerns the increasing prevalence of digital photography. As this BBC News article explains, digital photography may cause problems for future generations of local or family historians. Proper (printed) photographs tend to get stored away in shoeboxes in attics, and are still more-or-less as legible after a hundred years as they were when they were taken. Whereas an entire collection of digital photographs can be wiped out by one hard-disk failure.
Or maybe in fifty years' time nobody'll know how to display a JPEG. Somewhere I've got a tape with the very first program I wrote, recorded on it. It was a simple bullseye game for the Video Genie II (a TRS80 clone) - it wasn't particularly sophisticated by general standards, but not at all bad for an eight-year-old kid. I have no means of retrieving it on any of the small menagerie of computers currently in my house.
Another, related, example: in the mid-1980s a (for the time, pretty damned impressive) multimedia project was launched - the Domesday Project. This was a laserdisc containing digital reproductions of the original 11th-century Domesday Book itself (a census survey of the entirety of England ordered by William I) together with (I think) the 1981 national census data. All very innovative (albeit rather a costly system) but ironically, 15 years on, the laserdiscs are not readable by current technology - but the original 900-odd-year-old Domesday Book itself still is.
I guess the point is that it's all very well saying ``this media is guaranteed to last for fifty years'' (although personally I'd be happier with something that'll last several hundred) but also you have to guarantee that the data format itself is going to remain readily decodeable. This is not a problem for 1000-year-old documents on vellum (as long as your Latin or Norman French is ok, and you've done a basic course in paleography).
nicholas
This is an important question, and in my opinion has two particularly valid answers.
The first of these is the one that usually gets advanced - that (as with other pure scientific disciplines) we just don't know what `useless' knowledge might turn out to be useful or vital in fifty year's time. This is all well and good, and a perfectly decent reason to study something.
The other one, which I've come to believe more strongly over the past few years, is that which is often advanced in support of arts funding - that it benefits a society greatly (often in intangible and undefinable ways) to study and research things whether or not they have any practical use.
This is a point which, in the UK at least, a succession of education ministers have either missed or fundamentally disagreed with over the past few decades.
Last month, Charles Clarke, the current Secretary of State for Education made some very disturbing comments about how he didn't see the point in spending taxpayers' money on maintaining a group of ``mediaeval seekers after truth''.
He was initially misquoted as saying he didn't see the point in the study of mediaeval history, which rightly got a lot of historians angry, but a later statement clarified that he actually didn't see the point in studying any subject which didn't have a direct positive contribution to UK industrial or economic interests. Which I find even more disturbing - it's understandable (even ok) for the Chancellor of the Exchequer to have such a viewpoint, but I like to think that the Secretary for Education should at least see some worth in all of the education system.
A friend of mine (an eminent evolutionary and reproductive biologist who's also helped design aliens for people like Anne McCaffrey, Larry Niven and Jerry Pournelle, and co-written a couple of books with Terry Pratchett) once said
which I rather agree with. Studying or researching any subject changes the way you look at the world - often for the better. It teaches you new or variant modes of thought which you can then apply (often unconsciously) to other areas of interest.
For example: A former office-mate of mine now works for the NHS Breast Cancer Screening Service. The topic of her thesis (permutation group theory) is irrelevant to what she does now. But I find it tremendously reassuring to know that there are people that well-educated, and who have been trained to such a high level in thinking clearly and carefully, involved in something that important and worthwhile.
nicholas
I've also heard `MSRI' pronounced `emissary', which is perhaps a nicer way of looking at things - a bunch of people bringing news and information to a wider world.
On the other hand, though, there is a peculiarly masochistic element to research - especially in something as abstract as pure mathematics. It can be tremendously frustrating and misery-inducing at times, but when it's going ok it's great - in a strangely aesthetic sort of way. From what I've read of other artists talking about their work, it's not unlike writing, painting, sculpture, or composing in this regard.
Personally, I think it's really cool when a bit of mathematics I'm interested in turns out to have some applications in other areas of knowledge, or even in the `real world' (whatever that means). But that's not why I study the subject - I study it because I find it interesting for its own sake.
nicholas
I barely finished H.S. and I couldn't afford college. I learnt Calculus, Number Theory, Analytic Geometry, and Topology on my own. No college, no professors, no help, just an HP-48, pencil, and paper.
Wow. That takes some considerable motivation and commitment, and is an achievement worthy of respect. Congratulations!
I was worried when I originally wrote that post that it might sound a bit arrogant - I can only reiterate that it wasn't meant like that. This is, of course, a problem with text-based discussion forums - the finer nuances of the conversation can easily become lost. Sorry about that.
I should point out that I don't think that advanced mathematics is beyond an intelligent and motivated person - you yourself are a prime example of that (did I mention how impressive I think that is?). And I think that the majority of topics can be explained to the interested nonspecialist (New Scientist and Scientific American successfully do this all the time, for example).
But deeper and deeper levels of understanding require one to know about more and more subtle concepts, which requires more investment in terms of study. Some cutting-edge research is reasonably easy to explain to non-specialists. But some isn't - not because the researchers don't understand it well enough themselves, and not because the interested non-specialists aren't intelligent enough. It's because some of the concepts involved are extremely subtle and technical, and hence take some sort of advanced study (whether within a formal academic environment, or under ones own steam) before one can really get ones head around what it all means.
This, I suspect, is why there have been a couple of Slashdot discussions about the Riemann Hypothesis (because most computery types have some idea what a complex number is), or newspaper articles about the Fermat-Wiles Theorem or the Goldbach Conjecture (because both of those can be easily stated in terms that non-mathematicians can understand - everyone's been taught Pythagoras at some point, and everyone knows what a prime number is).
On the other hand, I've not seen any newspaper articles about the Hodge Conjecture, Yang-Mills Theory, or the Poincare Conjecture. Because it's difficult to explain to the general public, in a paragraph, what a `projective algebraic variety' or a `homotopy 3-sphere' is. And much less why they should care, or why some weird nutters get excited about them.
And that's a shame. Because those of us who do study mathematics to postgraduate level know it's a fascinating subject.
nicholas
Personally, I'm in favour of mandatory photo licenses for driving in the UK. It seems entirely reasonable that the old-style photo-less documents be phased out in favour of the new type. The old-style documents do date back a fair way, and society (especially the proportion of the population driving cars) and technology has moved on a little.
I am, however, quite opposed to the introduction of general ID cards for UK citizens, and wrote to the Home Office last week to outline my reasons for this.
The important distinction, you see, is that driving (despite what the majority of motorists seem to think) is not a right - it's a privilege (which should, in my opinion, be withdrawn from about 50% of current UK motorists).
I have various forms of photo-id of my own - a university library card (which I'm required to present whenever I want to borrow books from the university library), a railcard (which I'm required to present in order to claim a 33% student discount for UK rail travel), a passport (which enables me to travel to other countries ), an international student card (which secures me various extra privileges and discounts in participating shops and countries), and a university film society lifetime membership card (which lets me and a guest watch any film for free).
I don't, however, see why I should suddenly (or, more likely, gradually) be required to have an identity card in order to simply reside and go about my normal daily business, in the country that I (and at least several preceding generations of my family) was born in.
nicholas
If you want to try something of Eddings, beware. There are two 5-piece series and two 3-piece series.
Strictly speaking, there's just one series. He just keeps republishing it with different titles, and different names for some of the characters, that's all.
nicholas
So right now I'm sitting in a computer lab (working on an overdue assignment). There is a large sign posted (where everyone can see it) that says "No Cellphones!".
The problem is that most people mentally add the amendment ``... except for you'' to signs like this.
Thus, ``No parking'' becomes ``No parking except for you'', for example.
Virgin Trains, in the UK, started designating one carriage as a `Quiet Coach' in which mobile 'phones and personal stereos were prohibited.
Nobody takes the slightest notice.
I've stopped travelling in the quiet coach now, because I found that mobile 'phones ringing there irritate and distract me far more than they do if I'm in one of the normal coaches.
nicholas
countries like the EU states, China or Russia. Countries which definitely don't care about international problems and the mind-stunning threat of internation terrorism.
Without wishing to get into a political argument, as a citizen of a `second world' country, I'm a bit irked by your implication that nobody outside the USA cares about international terrorism. No disrespect intended to the thousands who died and were bereaved by the horrific attack on the WTC, but the rest of the world has been living with the effects of terrorism for a bit longer than the last year.
To take an illustrative example, over the last thirty years in the UK we've had actual elected Members of Parliament assassinated by both loyalist and republican Irish activists. Shopping centres have been bombed at Christmas, pubs blown up, ordinary people shot for associating with other ordinary people who happen to be `on the other side'.
I count myself fortunate that I've not been personally affected by such things, and I have every sympathy for those who have - like millions of people around the world I watched in horror as the WTC collapsed.
But to claim that countries and federations like the EU, China and Russia (which between them have roughly 1.5 thousand million inhabitants - a *quarter* of the world's population) just aren't interested in international terrorism is at once naive, insular, and offensive, and in the long run will only serve to fuel the sort of ill-feeling and fanaticism which causes the terrorism in the first place.
The world will be a mess and it will be because of whiners like you destroying the worldwide US surpremancy.
The world is a mess partly *because of* the USA's powerful influence. Don't get me wrong, there's a lot about the USA which I admire, and I don't think the UK has a particularly shining track record as far as international diplomacy goes, either, but simply dismissing everything as the other guy's problem is not the right way to go about making the world (or even the USA) a better place.
But then I'm a bleeding-heart liberal eco-head living in a `second world EU state', so what would I know?
nicholas
I'm glad someone mentioned G&T, and not just because my PhD supervisor is one of the managing editors, and my MSc supervisor is the other one :)
If I remember correctly, the whole thing was sparked off by Rob Kirby's article on the pricing of research journals. There's an interesting article by Joan Birman in the Notices of the AMS (vol 4, no. 7, Aug 2000, pp770-774) which discusses the various issues, and includes detailed discussion of the day-to-day overheads of running a free, properly peer-refereed research journal. It's available from her web page, in PostScript form.
G&T (and its sister journal Algebraic and Geometric Topology, and the related Monograph series) isn't some low-quality vanity-press thing - it's a real, proper, peer-refereed journal with high standards. At a quick glance, I recognise the names of three Fields medallists on the editorial board, as well as some other very eminent names in the field. And yet it's being run with virtually no overheads by two university lecturers (one of whom is semi-retired) in addition to their normal departmental duties (lecturing, administration, supervising research students).
I understand that a lot of the procedure is automated, with a mixture of TeX and Perl, with copies of all articles being submitted to the arXIv.
Ah yes, I'd almost forgotten about the arXiv. A central repository for research preprints in mathematics, physics, and computer science. It's an unrefereed archive for research announcements, preliminary reports, and preprints. Papers submitted to refereed journals often take up to a couple of years to actually appear in print, so the idea is that you issue a preliminary version of your paper to faster communicate your ideas to anyone else who might be interested.
This stuff is great - it's all about collaborative research and the free and efficient sharing of ideas, and it gives me a great sense of hope for the future.
-- nicholas
Thanks. Great explanation.
Very kind of you to say, thanks.
Could you elaborate and tie this in with the number of primes between m and n?
I'm a little less confident about this, but here goes...
As I understand it (and bear in mind that I've not done any complex analysis for several years, and number theory has never really been my forte) sometime during the 19th century Gauss noticed that the distribution of primes was approximated pretty well by a function he called the `logarithmic integral'.
Li(x) is defined as the integral from 0 to x of (1/t) dt. And apparently the number of primes below x (usually denoted pi(x)) is pretty well approximated by Li(x).
Now this is where I start to lose track of things.
As I understand it, it was proved at the beginning of the 20th century that the validity of the Riemann hypothesis is equivalent to the assertion that the deviation of Li(x) from the actual value of pi(x) is of the order of sqrt(x)*ln(x).
If I remember correctly, some work of the three great British mathematicians Hardy, Littlewood, and Hardy-Littlewood showed that pi(x) actually oscillates around Li(x) infinitely many times (although it really doesn't do it very quickly - the first value of x for which the graphs cross is very big indeed).
qv: the Clay Institute's page
and Chris Caldwell's page for better explanations.
As to whether there's a relatively simple proof out there, I don't know. My (non-specialist) suspicion is that there isn't, because some really clever people have tried to find one for a century and a half and failed. I'd be interested and impressed to be proved wrong on this, though.
Andrew Wiles' celebrated proof of Fermat's Last Theorem (if you haven't done so, read Simon Singh's book on the subject, and if possible watch the BBC Horizon documentary - the transcript is available) was pretty complicated and, I'm told (algebraic geometry isn't my field either - there's an awful lot of diverse mathematics out there) introduced some genuinely new ideas and methods.
The general feeling is that if there were a simple proof of either Fermat or Riemann (or, for that matter, the Goldbach or Poincare Conjectures) then someone would have found it by now - some really top brains have worked on all of these over the years (including several Fields medallists, FRSes, and the like).
(There's a guy who posts regularly to sci.math who reckons he's got a simple proof of Fermat which doesn't resort to all that scary stuff about elliptic curves or modular forms. The consensus seems to be that he's a nutter, though - his `proofs' contain obvious flaws which he refuses to acknowledge, claiming instead the existence of an enormous academic conspiracy against his work.)
It's also often the case (and this was true for the Fermat theorem) that proofs of such intractible problems, even those which are subtly flawed, introduce new ideas and methods of attack.
This is why otherwise sensible mathematicians have a go at these problems - even if they don't manage to solve them, the chances are that the attempt will inspire them to find new methods or potentially important partial results. Even had Wiles' original (flawed) proof turned out to be irrepairable, it was a pretty major piece of work which introduced some important new ideas which could well be useful in solving related problems.
My guess (as an interested non-specialist) is that while a proof of RH would be complicated and elegant, it would also involve some new twist or idea. As for who might do it, my money would be on Prof Louis de Branges of Purdue University - he demolished the (similarly intractible) Bieberbach Conjecture in the 1980s and thus seems to know what he's doing. Or it might be someone else entirely, someone who's spent seven years locked in their attic (as Wiles did).
nicholas