## Wednesday, December 12, 2007

### Prime decomposition fact! (And how the lecture went)

Before I give a narration of todays lecture by Dr. Walker, I want to make a remark! Every (positive?) number is either prime or not. Those that are not prime, can be written as a product of primes. i.e. the "Fundamental Theorem of Arithmetic". I realise now that I forgot its name today, because of the fundamental theorem of calculus! Darn that theorem...

I can only exist in two states (it seems): either bouncing about everywhere, or falling asleep in an empty classroom! This morning was a trek, but I managed to get to university in one piece. Sadly today was the final real analysis lecture, and I am too overwhelmed with emotion(!) to talk about the final week of term and the end of lectures. [It was another great lecture - the double dagger symbol was used as well!]

One thing I won't miss is the four people who sat in front of me in the vector calculus lecture. Have you watched Kill Bill 2? Do you recall a particularly gruesome scene involving the woman wearing the patch? If not then relax, but seriously my overactive imagination was going hyper of the (not) nice things which I could do. [This groups behaviour was disgraceful: they were talking loudly, tickling each other, behaving like you do in a zoo and taking pictures of ... GAH - they had no respect whatsoever for those sat around them and for the lecturer. Hence the Kill Bill movie moment in my head...]

That was the build up. At 12pm I became very nervous. "Are you coming to the lecture at 1pm?" "Ah, erm... nah... somethings come up." Fair enough unexpected things can pop up, but then why tell me that you will attend?! (Maybe to shut me up, but you could always say "Possibly" as one person did.) My friends and everyone told me to relax! I wasn't worrying about the audience for myself, but for the speaker. It is encouraging when an audience is present (or so I hope). Unable to sit still I made a nuisance out of myself, and sorted through a few things.

As 1pm approached only two people were in G209, which didn't help me. I seem to go into a different zones on lecture Wednesdays - a new level of bounciness is attained. It is excitement and nerves mixed with relief, and then just happiness as the scenes unfold - this causes a tremendous amount of energy to be produced, causing me to behave like normal (or to others un-normal?)

I had thought nothing of it at the time, but DC had remembered, and my words came back to haunt me. (Trust him to remember that!) Before I had to go and introduce Dr. Walker, I wasn't nervous whatsoever. I had a vague idea of what I would say (and I conspired with Fizz to try and embarrass Bella, which I successfully accomplished!) but I wasn't nervous.

There was an open day today and this caused a lot of confusion, for the room next to G209 had a lecture for future undergraduates. Some people had sat in the wrong room so I stationed myself outside, to direct people to the right room. For a second I became worried that PS might not be able to come, but was relieved when he walked towards the room! The audience had turned up. From two people we now had 40 odd people. I even nearly gave a false introduction at 1:05pm. (I can't tell the time it seems and had thought it was 1:10pm! Thanks to everyone, who in exasperation, pointed this out to me). Don't worry I am getting there, but I was suddenly filled with nerves as I tried to straighten my tongue for the introduction. Speaking to a smaller audience, I think, is more nerve racking that speaking to a crowd of say 100 people.

That is a rather naive thing to say, but forgetting what the fund. theorem of arithmetic was called made me panic. I need to take my time with things and not panic when there is silence for more than a second! Ah well, thanks to my silly proposition I will have plenty more times to get that tongue of mine straight.

Now for the lecture.

The lecture, I feel, did a great job of doing what a general audience lecture does. It gave us an insight into something, inviting us to ask further questions and read further. It began by setting the scene about symmetries. (This caused much distress to the numbers, sets and function lecturers for first years are taught to do compositions from left to right, whereas in the lecture they were done from right to left! DC only banged his head against the table a few times, so it didn't cause much damage...)

After this, the definition of groups was given. Thankfully (due to algebra lectures) I knew what was being said and followed the theory quite nicely. (New word for my dictionary: factor groups). The lecturer was basically setting the scene for everyone, especially for first year students who haven't come across groups yet. [Once again, the order in which I remember things is most likely wrong! Any corrections are welcome.]

Above I said that prime numbers are the building blocks for numbers; likewise simple groups are the building blocks for finite groups. That is, finite groups can be broken up into simple groups. A classification of simple finite groups was given, and it seems that if you want to make a name for yourself in maths, get a group named after yourself! (A theorem states that every simple finite group belongs to certain families: Lie groups, Sporadic groups, classical groups, alternating groups etc).

It is the Sporadic Groups which concern us; they contain the groups which basically don't belong to the other families. Eg. the monster group. The monster group is just a monster. I imagine it to be a big ugly looking thing, but the lecturer used more friendly adjectives. I don't know why but the idea of it being big and scary seems more appealing and complements the name rather nicely. The monster group is 19628 dimensional, so it deserves its name. There was mention of the j function and I became preoccupied for a second about why one term had no q. (It did register later but puzzled me for sometime).

Groups in general seem to play an important role to many other areas of maths. However, they are important in physics too! It is interesting to see how much certain fields are connected. The lecture was pretty informative and had a pleasant flow too it. There is normally a "register" of some sort going around (so we have an indication about attendance=>refreshments idea...). It seems that I had foolishly sent it around the wrong time and many people didn't sign it. Hence why I was straining my neck a few times to get a general idea about the attendance (annoying people in the process). This was silly of me, but we learn from such things.

An interesting thing about the alternating groups was mentioned. The symmetric group (S_n) can be divided into odd and even permutations. The group of even permutations is the alternating group, and they are simple for n \ge 5. There was a mention of these groups and about the general formula for fifth order polynomials. (Note to self: read up on this!)

Dr. Walker had a fantastic book with her: the ATLAS of Finite Group Representations, a nice pocket sized book which every mathematician carries. I must get myself one of them! I have probably written some mathematical mumbo jumbo, but I would very much like to know more about some things mentioned. The lecture gave a nice overview, and was very informative.

--

I would like to thank Dr. Walker for her great lecture, and to all those who attended. 45 people is just great, and please do attend future lectures. It is quite remarkable but two lecturers have approached me and volunteered to do a lecture! I hate telling them that the second semester is fully booked... if only there was a way, but the good new is that I now have six lecturers for next year. The bad news is that I don't have any students for next year, and if it remains that way, lecturers can always fill in. ;)

The refreshments were semi successful, since there were more people present than last time. (The jaffa cakes and pringles are what seem to be everyones favourites!) I would be interested to know how many people attended, having read my previous post. (One student gave me the impression of having done so... but I can't exactly go around asking random people about this!)

I must say that I am glad that the lecture series is over for this semester. As I saw my PT, the first thing that I commented on was about how exhausting it has all been. I really feel like I have ran two marathons, and am relieved for the small break now. Don't get me wrong - I have enjoyed every minute of it, and am pleased about how things have worked out. But that isn't going to help me pass any exams and get me through my second year. I fear my January exams for I feel that I don't deserve to do well in them, due to me neglecting my work. I can't get myself into work mode!

Let us not end the post on a depressing note. I hope that even when I leave, the Galois group continues to live. A tall order, but there is something about the community feeling that the society generates which is pleasing. Apart from me making a fool out of myself at the end, the interaction is great! Any feedback about the advertising etc would also be welcome, so that we can try to change things for the second semester. Hmmm, I think that's all!