MAST90053: Experimental Mathematics

2x assignments (worth 30% in total) and a take-home exam (worth 70%).

This is a difficult subject to review. To be blunt, I didn't particularly enjoy it, but I did learn quite a bit from it. Experimental maths is a fascinating area that most people never get exposed to, and this unit could potentially be excellent after a bit of a facelift.

The basic idea behind experimental mathematics is that exploring mathematical ideas with a computer can help you get insights that you wouldn't have been able to guess. After you make some mathematical conjecture based on the specific cases you've looked at using a computer, computational methods can also assist you in prove the general case. This general approach has been used to obtain a number of mathematical results in the last few decades that would have been essentially impossible without computational methods.

Unfortunately, the course doesn't really live up to the promise. For the first half of the subject, you spend each week looking at a different computational technique. Some of them are quite cool, e.g. chaos and bifurcation theory; the PSLQ algorithm for "number guessing"; Groebner basis. But the small amount of time devoted to them meant that you had almost but not quite understood how these methods worked before you moved on to the next topic, and certainly didn't have any opportunity to see where the applications of them might be.

The second half of the course I enjoyed a lot more. It was hypergeometric series, which are a very general form of infinite sum - if you've done any maths, you've probably encountered specific examples of hypergeometric series without knowing the name. There's no general formula for the sum of a hypergeometric series, but there are a number of computer algorithms for deriving a closed form for a particular series. This half of the course followed the textbook "A=B" almost verbatim. You will need that book.

While the unit outline mentioned computer-assisted methods of proof, there wasn't really anything in the course about it.

Dr Bedini's lectures were ... not the greatest. He spoke very quickly (while looking at the whiteboard, not at the class) and wrote a lot of stuff up on the whiteboard in borderline-illegible handwriting. To be fair, I think this was his first time teaching a postgraduate subject. But I eventually stopped turning up to the lectures, because I found that I didn't really understand the lectures unless I'd already read the relevant part of the textbook, and if I'd read the textbook, I didn't need the lectures. The lab sessions were devoted to implementing the methods from the week's lecture in Mathematica, and solving problems using them. In the lab sessions he was very willing to help out with anything that you didn't understand.

The assignments were an attempt to get us to actually discover some result we'd never seen before using experimental mathematics techniques, and then formally prove it. I think the second assignment, in particular, lived up to this ideal. The major assessment in this unit was a take-home exam. It was handed out at the start of the examination period (i.e. after SWOT Vac), and we had a week to work on it. It was 50% programming, 50% conventional mathematics. It took me a solid weekend's work to get it done, and I learned quite a bit while doing it. (I'd suggest that making sure you'd done all of the lab exercises beforehand would dramatically reduce the time you'd spent on it, because 90% of the programming question came from stuff I really should have already finished.)

One final problem with this unit was that, to really understand the methods presented in this subject, you need both a decent working knowledge of pure maths (specifically: ring theory) as well as programming skills. This was okay for me, since my undergraduate degree was in Pure Mathematics and I've spent years working as a programmer. But I suspect that quite a few people in the class were just left feeling perpetually lost.

No.

Dr Andrea Bedini.

No.

3.5/5

"A=B" by Petkovsek, Zilf and Zeilberger (1997). This is available as a PDF from the authors' website. Having a paper copy to refer to is nice, though.

1x 1-hour lecture, 1x 2-hour computer lab.

2014, Semester 1.

H1