Friday, October 24, 2008

A Hasty Mathematician

Yesterday was a Thursday and I had a Measure Theory lecture from 3-4pm. (Technically speaking it is 3-3:50pm, but that just doesn't look right!) I had mentioned that I was looking forward to the lecture as we would be continuing the proof of a Theorem stated on Tuesday. It was one of them "Andrew Wile's Fermat's Last Theorem Proof Lecture" if I may say so myself! The build up and excitement were all there, all we had to do was follow what was being said "without being expected to guess what was coming next". That sounded like a good plan to me....

NOTE: I will edit this post tomorrow with the maths symbols inserted, so please endure the TeXing. (I'm feeling too lazy today...)
The Theorem was that M - the set of all Lebesgue measurable subsets of \mathbb{R} was a \sigma - algebra. [This means that we have to prove (1) that \emptyset \in M; (2) If E \in M then its complement is in M; (3) If {E_j}_{j=1}^{\ifty} \contained M then the union of the E_j's is in M].

Proving (1) and (2) was somewhat trivial (two lines only...), but proving (3) is the fun part. We first consider two sets E_1, E_2 \in M and show that E_1 U E_2 \in M. To do this we have been given this sparkling, out of the box definition and we don't know how it works, but it works which is what matters! (The definition is about when a set E \contained in \mathbb{R} is said to be measurable).

I shan't be throwing any equations into this post, but consider this a preview of what's "coming up" (soon....).

So we first started by proving that the union of two sets is measurable. This seemed to take a while, but I was following the game so that was nice. However then something bad happened. Prof S' made the mistake of saying "Aha". The "aha" had been said. I hurriedly wrote the last line of the proof and drew my black square (as always) to signal the end of proceedings. I even took a moment to flex my muscles and prep myself for the remainder of the lecture.

How very silly I had been, but I will cover my naiveness by blaming the false "aha". (In my book they are strictly relegated and said at the end of proofs). We had proven the case for two sets (and an induction argument can extend this to n sets), but that's not what the third condition of a \sigma algebra is! The "note" that followed my black square didn't make me realise that I was being very silly, but when Prof. S' said that we now suppose that we have a countable collection of sets, I felt the egg on my face. Quickly crossing my small black square out, I pretended as if nothing had just happened.

This proof had just got better - Andrew Wiles eat your heart out.

So we continued with the proof and I wondered whether we would complete it by the end of the lecture, as I couldn't begin to start planning my weekend otherwise! I got lost somewhere in the argument of the next part but quickly found my feet again. I don't think that I would have pointed out the missing lambda to Prof. S' had it not been for the student sitting on my right. The student had not noticed it missing and her neighbour too had not noticed anything amiss, which was all I needed to open my big mouth.

The lecture was drawing to an end and we only had one inequality. Internally I knew that this proof would continue onto Tuesday, because to prove equality we have to prove that the inequality holds in both directions.

But then the tides changed direction: Prof. S' seemed to be smiling! He also looked to be saying that we only needed to prove the inequality in one direction to prove that the countable union is measurable. Where he got that from I had no idea, but hey, the proof was complete! A cheer ran through the room (although perhaps only near where I was sat...), but nevertheless the delight on everyone's face was undeniable. Prof. S' was so taken back with the atmosphere of the room that he awarded the proof with a very big black square of his own, which was a treat in itself!

Limbs were now stretched and smiles were plastered on everyone's faces - we had completed the proof in two days!

I must say, for those who might think that they're detecting sarcasm, there is none in the above. I was honestly really into the lecture and it was quite a sense of achievement when the big black square was drawn at the end.

This takes us to my next topic: Prof. S's lectures. Prof. S' is a great lecturer. Honestly speaking he's been telling a brilliant story about Measure Theory so far, and I know that it will remain this way until I have to read up on chapter 2 by myself! He's quite a brilliant story teller and after one comment he made about definitions and tea, my friends seemed to think that it came out of my mouth! (Although mention of tea did score him extra points). His lectures have a certain "casualness" and "mathematical drive" at the same time which is quite cool. I have yet to leave planet earth for long periods of time during his lectures, and even when I'm having a long and tiring day, I am alert for Measure Theory.

Once again, boy am I glad to have done Measure Theory this year, and if I'm lucky Prof. S' might be lecturing me next year too.

By the way - I did it again - I asked advice because I wanted an accomplice. I'm still not convinced about how far I want to take my mathematical education, but the fourth year has to be a must now.

I had left at the end of the lecture but sneakily sneaked back to the example class in the last five minutes, to enquire about the inequality of the proof. (My notes are disjoint and I had misplaced one set of lecture notes which answered my question.)

After this I spoke to Prof. S' about TGG and other things. He's quite a busy man it seems and has been an unbelievable help for TGG (he saved me from my despairs last week). He also might possibly be able to help me with a speaker for the second semester, but most importantly I feel that someone's working with me and not against me. He kindly shattered my bubble about being clever and booking rooms for next semester now, as it is not allowed. Teaching rooms for next semester timetable have to be booked first and then it's our turn. Nevertheless, hopefully this means that other "people" can't also book rooms and I can just play around with the dates I want to hold the lectures. (STUDENTS: Please volunteer to give a lecture. Don't think twice about it, just nod your heads and go for it!)

Go for it! Ha - that's exactly what I was told to do when discussing my silly indecisiveness about doing four years or more. We talked about quite a few many things, and then I finally asked Prof. S' about dynamical systems. (You see the dynamical systems seminars are on Wednesdays and were part of this week 7 clash). Going by Lagrange's definition yesterday, I would say that Prof. S''s "has thoroughly understood his work" (!) but then again I am not exactly an ordinary person on the street... Nevertheless, as I am exploring areas of maths which might interest me, dynamical systems has me asking more questions. It wasn't just mention of chaos which had me going wild (!), but the subject itself and how it's studied is quite interesting.

By the way, I officially hate the word colloquia for I always end up saying "British" before it. Pfft.

To conclude this post, I hereby remove the invisible clause which meant that Prof. S' had to attend all TGG lectures organised, for the rest of his life. (I would have told him the "rest of his life" part in due course....). He can attend whenever he wants to although I would obviously be a happier bean if he attended most times! This doesn't mean that all the other lecturers are off the hook though!

PS: The Topology lecture today happened to be a blur, but I was awake enough to hear Dr. E mention my name during the lecture. Now don't be hasty and assume that this bean has got a fantastic connection with Topology, resulting in a startling powerful discovery which Dr. E was exclaiming about. Alas no. Instead Dr. E was pointing out that he wasn't numbering something because "beans doesn't like it when I number things in this way". He just wanted to send the angry mass of students to me instead. Hmmm - second blood has been drawn, and this battle is going to get bloodier! Children - stay away from Topology, it's not the safest of subjects and believe me on that...

I drew first blood the other day (unintentionally of course) but my next move remains to be seen. Ideas are obviously welcome, but do keep in mind that Dr. E is a topologist. :D

If you have only just started to worry, or have just put the phone down after ringing the people in white coats, I can assure you that I will try to make my next move "mathematically painful". Happy? Alas, I have had a very busy week and need to sleep now, hence this nonsensical mess. You might be relieved to hear that I will be without the internet soon, and my blogging will most likely happen from the University cluster until rearrangements are made.

I thought this post had been very long so had published it but only to find it not very long. Now it should be very long and waffly so my work is done. Here's hoping for a mathematical weekend. (Now, now - don't laugh please! We can all live in hope...).

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