And then there are the depressing stretches. Many students seem incapable of writing a single correct sentence in their mother tongue even before your start looking at the maths. Looking through those can be an effort – not just because my obsessive-compulsive self has to restrain itself to not cross all that gibberish out, but also because it makes it much harder to figure out what people did or at least tried to do.

Figuring out what people are doing there is also much harder when they don’t bother to write anything but formulae. While I don’t require them to write essays, a few short notes would certainly help: Writing what they are claiming to be true, stating where the proof starts, stating that they are using Induction as a method, clearly marking where they use the Induction Hypothesis and so on. None of these is hard to do and they have been demonstrated many times throughout the term. But still, writing things up properly hasn’t made it into the DNA of the students after a term.

Admittedly, getting used to writing maths properly is one of the bigger challenges when starting to do it at uni as school ‘teaches’ people mainly to do computations and usually omits the main point which is making an argument – i.e. pretty much the opposite of what people should be learning and (attention overly optimistic world-view!) what people might enjoy more. I quite liked the system we used for freshers when I was at Warwick: In addition to the normal marks for exercises, the students could be awarded ‘clarity’ marks for their writeup. This was great as on the one hand marks speak a very clear language that students can grasp more easily than well-intended comments by their supervisors. And on the other hand it was entirely positive. They could get full marks without any of those ‘clarity’ marks (which was very and and unlikely in most cases) or they could sit down and think a bit about what they write and be rewarded it with this bonus. I think that worked quite well. The students’ way of writing things up improved and as a consequence their work also became better and easier to read and mark. A win for everybody.

But even apart from the writing, many students have problems with solving the (at most moderately difficult) exercises correctly. And a few hand in tragic work. When you read it, you get the impression that they either never visited lectures or everything taught there didn’t stick at all. Their answers just seem to be a collection of factoids stuck to one another in no sensible order and with no idea what’s actually going on. When reading their answers you start wondering why they didn’t run out of the exam, crying. It’s almost tragic.

Luckily, every now and again there will be a really nice set of answers which gets things just right: All the text necessary to explain things but no more. A clear argument, written up with linebreaks and white space to make it jump at you with no extra effort. Those were rare but could cheer me up again.

Which brings me to the topic of odd marking techniques. Of course you need to mark things in a way that is fair and can be understood. But when looking at the less thrilling work handed in you start observing a few things: With a few basic exercises to solve, the marks people get for all those are correlated. You could try to play statistics on those to still get a reasonable result. But of course that’s not an option. And, interestingly, after a few dozen samples, you’ll probably have a fairly good idea of the marks the student is going to get after reading the first line of the answer. I assume the style of writing is related to how secure and relaxed students are which again is related to how well they know the topic. Of course things can still go wrong after a start that looks good and very rarely there are positive surprises after a not-so-promising start, but the general trend seems to be there.

The final issue is writing. Girls’ work is more pleasant to mark simply because it’s usually more legible. A few students had prepared their own exam ‘stationery’ for which they had printed their name and uni number on each page to save the time/effort of having to write it themselves in the exam. A sweet idea, I guess, even though using Times New Roman for it isn’t the most stylish thing you can do. Still beats the person who used Comic Sans, though. I really had to discipline myself to not deduct marks for lack of style there…

Essentially you start with a circle and put n points on it. Then you join each point to all other points using straight lines. The question is how many areas this divides the circle’s interior into for any given n. And to give an argument for the answer you find, obviously.

That question can be rather hard to answer and we’ll add one more provision to it: You can assume that no three lines meet in a single point. E.g., for n=6, if the six points are arranged as the vertices of a regular hexagon inscribed in the circle, three of the hexagon’s ‘diagonals’ would meet right in the centre of the circle. That is a bit of a ‘singular’ situation and essentially removes a triangle – the one with parts of those three lines as its sides – from the count. Which messes things up. As we want a definite number as a result, we disallow such triple intersections and go for slightly less symmetric point arrangements.

It may be interesting to think about whether this additional provision actually makes sense. It’s obvious how we can move things just a bit in the case of six points. But will this still work when there are sixty, six hundred or six million points? Luckily this technique of moving things a little to avoid the undesired situations will work for arbitrarily large numbers of points thanks to Sard’s theorem. Which means that hiding in this seemingly harmless provision is a pointer to yet more maths.

As the exercise was just a fun and voluntary bonus one, people didn’t actually do it. Hence I rest my hopes on you to give the correct number and argument…

Only later on, you start appreciating Taylor series. Possibly in complex analysis where they gain a lot of power and simplify things a lot or, for the more practically minded, because Taylor’s theorem gives you a way to approximate things and get a bound for the error. Physicists are particularly notorious for saying things like using the first two terms of the Taylor series will give a precise result. Which is totally in-character, as the statement is usually blatantly wrong, but still true for all practical purposes.

The most famous Taylor series will be that of the exponential function, developed at 0. Use it for the value 1 and you can approximate Euler’s number. It’s surprisingly easy to do and converges surprisingly quickly.

I.e. the first five terms already give 2,5+ 0,16̅ + 0,0416̅ = 2,7083̅ with the remaining error being smaller than 1/120. Not too shabby.

To perhaps make students think about this and its relevance a bit more, I thought I should point out that if their calculators or computers need to compute trigonometric functions, Taylor series expansions are a way they can achieve that magic. And then I started wondering whether in practice they actually do so. Now that was a bit tricky to find out.

XCode’s help didn’t mention how sin and cos are implemented on the Mac. The best I could find there was a note on Mac OS X PowerPC numerics. The document contains choice passages like this one:

The cosine, sine, and tangent functions use an argument reduction based on the remainder function (see page 5-11) and the constant pi, where pi is the nearest approximation of π with 53 bits of precision. The cosine, sine, and tangent functions are periodic with respect to the constant pi, so their periods are different from their mathematical counterparts and diverge from their counterparts when their arguments become very large.

And that strongly suggests that no Taylor series are used there. In fact, the whole thing is based on a constant π. Which of course is wrong because computer architectures aren’t good at storing irrational numbers. Eeek! Another document on the web (link, as well as many of the others, kindly provided by Tom) suggests that the problem isn’t really PPC specific and that people found Java’s performance ‘bad’ on ‘x87’ because the virtual machine tries to avoid that problem. Amusing quote from that text:

So Java is accurate, but slower. I’ve never been a fan of “fast, but wrong” when “wrong” is roughly random(). Benchmarks rarely test accuracy. “double sin(double theta) { return 0; }” would be a great benchmark-compatible implementation of sin(). For large values of theta, 0 would be arguably more accurate since the absolute error is never greater than 1. fsin/fcos can have absolute errors as large as 2 (correct answer=1; returned result=-1).

While Java might do better because of moving all arguments for these functions in the [0, 2π[ range, this suggests it still doesn’t necessarily use Taylor series. The next question would be whether or not Taylor series expansions actually make sense performance-wise for computers or whether programmers came up with more efficient algorithms for their limited usage cases.

All this still doesn’t make a load of sense to me. Most likely because I’m not a programmer and because strategies for implementing such a non-trivial computations can vary wildly with the times, fashion, hardware they run on and the program they run in. Now I’ve become curious and I’d like to know how various processors and maths libraries compute their trigonometry. So what about making a little table with results for sin(1), sin(100000) and sin(100000000) run on different systems? In fact, we started making one already which will appear in the next post. Be sure to join in and provide more examples – particularly if you have exotic hardware.

Bonus material

• Comments in the dosincos.c code of glibc suggests it uses Taylor series in some implementation. Seems to be some strange vector code(?) So some machines may actually used the beloved and correct series?
• Odd but fast implementation
• Avoiding sines when drawing circles for better performance in retro computing
• Intel maths libraries
• A paper in trigonometry implemention on AMD (€€€)

I put the main blame for this on the fact that linear maps from a field to itself are nothing but multiplication by a number of the field. And this set of linear maps is the field itself. In a way I am tempted to put the blame for student not fully understanding derivatives on the isomorphism

which is hiding everywhere. You tell your freshers the definition of a derivative and you also tell them that it’s a linear approximation of the function at that point. But give them a week and they’ll have forgotten about all that linear nonsense and just think about a derivative as just another function that spits out numbers. However, these numbers are really to be thought of as linear maps, i.e. you get a linear map for each point of the function’s domain. That linear map just happens to be writable as a number. But people need to reach their second semester and learn about multi-dimensional differentiation to really see that.

This misconception really hurts when you look at the chain rule for taking derivatives. In the one dimensional case it is tempting (and working) to think of it in terms of ‘inner derivative times outer derivative’. However, keeping in mind the idea of linear maps, it may be conceptually better to think of the situation as follows: at each point of the domain you get a number which corresponds to the linear map that is the composition of the linear map which is the derivative of the ‘outer’ function at the result of the inner function of that point composed with the derivative of the ‘inner’ function at that point.

In formulas this means that conceptually you see better whats going on if you don’t write (f∘g)′(x) = f′(g(x))g′(x) but instead write D_x(f∘g) = D_g(x)f ∘ D_xg.

Of course it seems a bit crazy to use all those complicated words and notation for something that when doing the computations in a one dimensional exercise boils down to multiplying two numbers. It’s not that the simplified version is wrong. However, the ‘right’ idea, the idea that can be generalised to higher dimensional cases will remain hidden when using the simplified notation.

The ‘better’ notation also highlights that it’s really a good idea to not think about a derivative as a function which you plug points of the domain into. It’s more a collection of linear maps for each point of the domain. That could be another rant…

I’m not sure whether they actually revised, but at least the stress aspect worked. I was quite surprised to see people turn up for this with loads of pens and mountains of chocolate to get themselves through those thirty minutes. I was even more surprised that there are people who are on their second or third attempt to pass the course and who were really stressed by this. Even if maths is just their second subject for becoming a teacher, how can they possibly come out of this as a good teacher? Won’t they become the teachers who ruin maths for their pupils?

With all the ‘stress’ the irony is of course that classes (and probably any uni course) are operated on a ‘Don’t be Evil’ policy. Nobody wants the students to fail and they get all the help they want if only they care. And at least the students I am teaching heard me tell them quite explicitly that they should really know this or that definition when exams are coming. Most of them took the hint, it seems. Just one of them let me know that he didn’t but he revised them after the test. Which, I suppose, does the job as well.

When all students get the same assignment to do in the undergraduate days, all of them will have to access pretty much the same books in a short period of time. As there aren’t enough of these books in the library to let each of them have one, there can be a bit of a run for the relevant works. And soon someone figures out that he can simple wrongly re-shelve a relevant book and thus get a competitive advantage. Both by having the book and by depriving his fellow students. The book will still be accessible to him the next day but it will be lost to everybody else. Some call that clever, I’ll call it anti-social. And surely people who do it should be kicked out of the library or university.

While maths departments not quite wrongly have the reputation of being a bit of a freak-show, a good thing there is that people are generally decent and simply wouldn’t do such things. Or would they? Well, this week I needed a book and couldn’t find it where it was supposed to be. By sheer coincidence a friend saw it sitting on a completely wrong shelf (not accidentally mis-placed but neatly put in a ‘secret’ stash with another mis-placed book) and thus I got it after all.

However, I was not impressed.

So – for mathematicians that’s three times two, while for physicists it’s more like ten minus four.

Obvious, but in that form qualifying for quote of the week, I’d say.

So I was a bit sceptical when I saw the speaker stand there with his computer today. But it was encouraging to hear him indicate right at the beginning that he is aware of the problems of slideware talks but he had seen one which didn’t make him fall asleep last year, so he wanted to try that as well (for his first talk, not for the following ones where the real work is done). Nice joke. And also indicating that he had thought about the potential pitfalls of this approach.

The slides ended up being relatively simple with just a few words and possibly formulas each. Sometimes containing a shrunk version of a key aspect of the previous slide in some corner for better reference. Not graphically perfect but getting the job done and with numerous interesting ideas.

That said, seeing the slideshow made me realise that as soon as people have something non-trivial to say, those techniques of pretty photos and nothingness aren’t worth much (and while I’ll salivate over such slides for sure, how exactly do they improve an argument in a way that the speaker couldn’t?) just don’t cut it. Probably the difference between people who sell and people who make an argument.

It was also the first maths presentation I saw being done in Keynote. I suppose that contributed to making the slides a bit more lively as it opened the way to doing things that T_EX can’t do while still being open to displaying (what looked to me like) properly T_EXed formulas and diagrams — thanks to its good PDF integration I suppose.

But as the speaker pointed out: this was just an overview talk without the technical details. Once the details come, there’ll be quality blackboard time again. Possibly because the mostly static digital presentation doesn’t cut it for them.

Finally I also wondered about graphics. Unless you are doing the ‘pretty photos and nothingness’ thing as mentioned above where the actual graphic you have doesn’t matter too much and some stock photo seller is bound to have something appropriate, graphics are hard and extremely time consuming when wanted in a digital presentation. Say you want to draw a picture of a torus with a closed loop in it. You can draw that on a blackboard in ten seconds without a problem. Sure the torus will not be perfect, but it will do the job just fine. Now compare that to the digital situation: In the digital situation having a graphic where lines don’t match up precisely and smoothly will immediately look broken next to all the perfectly rendered text. Of course you’ll also need your torus to be nice and symmetric. Now think which tools and amount of experience with computer drawing tools you’ll need to achieve those results… and whether your average mathematician has the expertise or budget to get those good graphics.

Now they changed things. And all lectures start and end on even hours. The lunch break will be a matter of luck, I suppose. I find this quite odd. Usually the university is pretty immune to change and I wonder why that happened. Perhaps because there tend to be more and more students and so they have to cram more stuff in the same time frame (although there isn’t more staff, so it’s not quite clear who should actually teach the extra lectures). Or – not that it is particularly common in our department, but it is over at the philosopher’s – they want to make sure that all lectures and seminars are over at six in the afternoon. So they can turn off the light and heating to save money.

The thing about light and heating certainly isn’t a joke. Apparently the university is so broke that they urged all the departments to completely close down between christmas and the new year so they can save money by closing down the buildings for a week. Sad.

Today was a ‘big day’ in mathematics because the ICM opened in Madrid. It takes place once in four years and it is the occasion on which the Fields Medals (aka ‘maths Nobel prizes’) are awarded. Being the masters of drama that they are, the mathematicians decided to award the prizes right at the beginning, so they can move on and do serious stuff for the rest of the week.

Four Fields medals were awarded to Andrei Okounkov, Terence Tao, Wendelin Werner and Grisha Perelman. To be honest I didn’t even know about the first three, but Perelman has been well known since 2002 when he put a few preprints on the arXiv which were said to essentially prove Thurston’s Geometrisation conjecture and – as a consequence of that – the famous Poincaré conjecture. I think pretty much right away everybody who is proficient in that area of mathematics and could understand the arguments quickly was amazed and pretty confident that the techniques he used will actually do the job. Apparently it was really solid work.

And thus, he was a pretty hot candidate to be awarded a Fields medal. Not only does it look like he has solved a fiendishly hard problem, but he also turned 40 this year – the maximum age at which you can receive a Fields medal. Actually he was such a hot candidate that even the mainstream press managed to write about in advance. Of course the didn’t use the opportunity to say what Perelman prove but rather used choice quotes from other mathematicians like It’s a central problem both in maths and physics because it seeks to understand what the shape of the universe can be. (so what exactly is that supposed to tell us?!) and elaborate on the freakiness of Perelman as a person. While I don’t think that it’d be strongly misleading to portray the world of mathematics as something resembling a freak show, it still seems inappropriate to me to report the social skills – or lack of – of the protagonists in a way that exceeds the reporting of what they actually did and may make people dismiss the whole achievement because the people in question are odd.

One thing that makes Perelman newsworthy is that he declined accepting the Fields Medal and likewise he didn’t make an effort to turn the solution he came up with into a ‘proper’ paper with all the details worked out which can be published in a journal. Doing this could be worth a million Dollars as the Poincaré conjecture is one of the milennium problems set out by the Clay Maths Institute. But Perelman doesn’t seem to be interested in that either.

So what is the Poincaré conjecture, you may ask – a point that seems to be mostly ignored by the mainstream media. The fun thing is that it can be stated in a short sentence. The not-so-fun thing is that there are some non-trivial words in the sentence Every simply connected compact 3-dimensional manifold is homeomorphic to the 3-sphere. I’ll try to explain some of these words and make a few remarks on them. Let’s start with the easy bits:

manifold

That’s maths talk for ‘object’. Err, that didn’t help! So what’s an object then? A reasonably well behaved shape. Like a point. Or a line. Or a circle. Or a sphere. Or a torus, to name the favourites. Those are simple examples. A manifold has a defined dimension n (i.e. you can’t have something like a sphere and a line together) and indeed once you ‘zoom in’ on it, it looks like the well-known n-dimensional real space everywhere. E.g. just looking at a part of a circle will give you a line which in turn is 1-dimensional real space.

homeomorphic

But stop you say, the little bit of the circle is round while a line is straight, so they aren’t the same. That’s what it looks like for sure. Particularly as I didn’t mention yet that we are doing topology, aka bendy mathematics, where we take the liberty to consider all things ‘the same’ which differ from one another only by bending and squeezing (but not by tearing!). To avoid having to say ‘the same’, there’s a special word for that which is ‘homeomorphic’. A round line is homeomorphic to a straight line and – the mathematicians’ favourite example – a doughnut is homeomorphic to a coffee cup.

3-sphere

That’s a 3-dimensional sphere. It’s a bit hard to imagine, though. Just consider the 2-dimensional sphere (e.g. a football): While it is only 2 dimensional, you cannot place a 2-sphere into a plane. It just won’t fit in. That’s why it’s hard for us to imagine a 3-sphere. But luckily there are a number of tricks to think about it. One is to start off with a 3-ball (i.e. a 2-sphere that is ‘filled’ which makes it 3-dimensional). And then you use the 2-sphere that is the ball’s outside and glue it all together into a single point. Well, you can’t really picture that either, but you can do it for the 2-sphere (start with a circle, which is the outside of a 2-disk, then lift that circle up and shrink it down to a single point, voilà you’ve got a sphere) and then at least get a good technical idea about how things work in higher dimensions.

compact

Let’s just say that means ‘small’. Not in the everyday sense of size, though, as we can stretch things arbitrarily but in the sense that it’s not infinitely large. For example the line of real numbers would not be compact but the circle is.

simply connected

A manifold being simply connected means that if you have any circle in the manifold you will be able to contract (not cut or tear!) it to a single point. This is a great way to detect holes. For example, the 2-sphere is simply connected (just try playing with an elasting band on the surface of a football) but the torus isn’t simply connected (when the circle goes around a ‘hole’ of the torus there’s no way to contract it).

All right, I hope that wasn’t too bad.

The key point there somehow is the following: It turns out that there are loads of manifolds and that many of them are homeomorphic. But actually proving that they are can be quite a pain. So people came up with really clever ideas (homotopy and homology, say) which let you reduce the things you have to look at to much fewer and technically easier situations, e.g. by looking at some properties of circles which you can embed in your manifold. This is extremely powerful. But it simplifies things slightly and it can happen that manifolds look ‘the same’ from the point of view of these techniques and yet they aren’t homeomorphic. And for the special case of simply connected compact 3-dimensional manifolds, the Poincaré conjecture asserts that we don’t need to fear: Whatever looks reasonably like a 3-sphere actually is a 3-sphere.

The strange thing about the Poincaré conjecture is that in the century of its existence many topologists gave it a try. And each of them failed and gave up after some years. It looks like a reasonably easy problem but it proved to be so subtle that it defied being proven.

And in the proof – if we may call it a proof already – we have now goes beyond topology. In fact, Perelman proved Thurston’s Geometrisation Conjecture which is a much more technically difficult conjecture that classifies the geometrical building blocks of three manifolds. Here ‘geometrical’ means that unlike in topology things like size and things not being straight do matter. Having this should be a great help for three dimensional geometry.

Further Reading

• ICM press text
• Clay Math Millennium Problem text
• New York Times Article
• Perelman at Wikipedia with plenty of links
• Poincaré conjecture at Wikipedia
• Thurston’s geometrisation conjecture at Wikipedia
• slightly ranty and outdated text on the Poincaré conjecture illustrating the history and frustration a bit