I’ve been marking homework for Group Theory. (Another exciting Sunday afternoon for me, eh?) I was reminded of a question last year from a student. I was discussing mathematics with some students and one asked me about showing a group is abelian. Their homework question asked them to state whether a particular group was abelian or not. The student liked this question because, she said, it didn’t ask her to justify her answer. She had a 50/50 chance of getting it right and that was good — she knew that she would be unable to supply any reason if asked and so would stand no chance of getting marks At first, I mistakenly thought she we wanted me to explain how to show a group is abelian. I was wrong. What she wanted was a shortcut, some rule of thumb say, that allowed her to tip the chances in her favour, maybe 60/40 or 70/30. Now, I’m not sure if there is a decent rule of thumb (if you see Sn or Dn it’s not abelian maybe) but this is definitely the wrong way to go about getting a good degree. Trying to memorize all the short cuts without understanding is not a good recipe for success at university-level mathematics — even if that strategy worked reasonably well for A-Level! The message for any student reading this is simple. If you have a weakness in your understanding — the student above certainly knew she did — then aim to remove it not by using some shortcut but by understanding the problem. Shortcuts will help only in the short run. In the long run they will do more damage than...

## Milnor wins Abel Prize...

posted by Kevin Houston

As you may know, I’m a big fan of John Milnor, I posted about his video on D’Arcy Thompson. Well, today he won the Abel Prize. My own research has been greatly influenced by Milnor. His seminal work on the topology of complex hypersurface singularities is the foundation for the work in my PhD thesis. What is amazing is that he seems to have come into the subject in the late 60s, knocked out a book on the subject and then left the field to do something else groundbreaking. He was great at exposition. Really great. His early 60s book on Morse Theory has not been surpassed in the field for its clarity and choice of material. For me it’s one of the great books of the 20th century. It’s up there with Orwell’s 1984. As a postgraduate I read his book on the h-cobordism theorem when I was on holiday. How nerdy am I? (It’s also a great book but a bit more challenging shall we say.) More information about his achievements, certainly more than I can fit here, can be found at the official winner’s page:...

## Fourier series cartoon...

posted by Kevin Houston

A cartoon in the New Yorker: http://www.newyorker.com/humor/issuecartoons/2010/10/04/cartoons#slide=6

## Pi is 4 video

posted by Kevin Houston

I mentioned the “proof” of π=4 in a previous post. In time for Pi Day on Monday, I’ve created a new video setting out the problem. For those not inclined to view videos here is the problem: Take a circle of diameter 1. Its circumference is π since its radius is 1/2 and a circumference is 2π times radius. Now put a square round it. The length of the perimeter of the square is 4 since each side has length equal to the diameter of the circle. Now fold in the corners like so. Since there was no stretching or shrinking, the length of this new curve is also 4. Do the process again, i.e., fold in all the corners. The length of this curve is still 4 since no stretching or shrinking was involved. Do it again. The length is again 4. We can take the limit of this process. The limit is a circle. Since the jagged curve gets closer and closer to the circle and always has length 4 we can see that the perimeter of the circle has length 4. But the perimeter length is also equal to π. Therefore, π is 4. Where is the...

## Millennium Maths: Hodge Conjecture...

posted by Kevin Houston

Another article by Matt Parker on the Millennium prize problems worth a million dollars each. Never mind solving it, this one, the Hodge Conjecture, is a bit tricky to explain in everyday terms (I know, I’ve tried):...