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DCG and -K

November 26, 2010 Leave a comment

I cannot wait for next Wed!! Once that’s over and done with, and I’m still alive, work can commence! The REAL work (for the mathematicians, please.. no pun intended)

More details coming up!

Falcon, OUT!!

Categories: Happenings

Ee-zee

November 19, 2010 2 comments

Consider this: You’re in Singapore, taking out your ez-link card, tapping the thing on the bus. Soon after, you accidentally grip the card too hard, it breaks into two.

Sigh. OUT!

Categories: Uncategorized

Th3 J0uRn3Y so far…

November 15, 2010 Leave a comment

While waiting for the bus, I sat there contemplating how my journey in Math has been so far.. Here’s what I got..

In Year 1, I came in, learning the idea of rigor in Math, how we need to prove things and not just use intuition. I would say that the first year was the beginning of struggles for me. I did pretty well, I would say, and by pretty well I mean above average. It was then that I enjoyed the simple, innocent pleasures in Math, thinking about trivial matters and building up a solid enough foundation for the years to come.

Year 2 was, I’d say, the steepest learning curve in my Mathematical journey. It was during this year that I matured a lot mathematically speaking. Two things – Linear Algebra II S version and my UROPS – were the highlights of that year. With Linear Algebra II, I was lost in the abstractness of things that I thought that as long as it works abstractly, I’m alright. Boy, was I wrong. Fatally wrong. Needless to say, I was punished for that belief, struggling the entire semester through… Didn’t know much of what actually happened until very much later… My UROPS, was, well, my first opportunity to get a feel of what mathematical research is like. Mathematically naive as I was, I don’t think it went really well. Looking back at it, I don’t think that I have learned as much back then than when I reflect on it now.

I’m not sure exactly when I found out about Terence Tao’s blog. It was probably during Sem2 of Year2 or Sem1 of Year3 or somewhere in between, but when I did, it opened my eyes to many new things. I read a lot on what learning Math is all about, how we need to internalize things, how we need to get the feel of things, understand the bigger picture and not to get lost in the rigor of things, simply put – “don’t miss the forest for the trees.” The reason why I think I didn’t learn much in my UROPS was because I focused too much on the details. Same reason why I didn’t really do well for LA2. When I realized this, I tried to switch my focus in Year 3 from being too focused on proofs and rigor to understanding the bigger picture, getting a feel and an intuition about how things are. It didn’t really work out, but trying to do it worked wonders for me. I began to realize how doing Math that way made not only solving problems easier, but learning more interesting. It was probably the reason why I didn’t really do well for my exams back in Years 1 and 2 – well, actually in Year 1 I did OK because we just needed simple tricks to make things work – there wasn’t much emphasis on ideas and intuition. Year 2 was when I was punished for that.

In Sem 2 of Year 3 I learned about how intuition and motivation is very important in studying Math. I was fortunate to have an Olympiad Silver Medalist in my Representation Theory class. The questions he asked opened my eyes to the things we can learned if only we bothered to explore further and ask about initial motivations and intuitions. Often times we learn interesting results, and we learn the proofs, but really, sometimes it leaves us wondering, how on earth did these imba people come up with these things in the first place?! They’re the stuff of amusement for most parts of our casual Math conversations, but a huge part of Math comes from thinking about how and why they could think about such things. Ramanujan, for example, had a list of crazy (but true) identities in his notebooks. My supervisor, with his supervisor, had proven some of these identities, and they are beautiful identities, but he still wonders how Ramanujan actually came up with these things in the first place – the general mathematical community are equally baffled by this – the fact that we have such modern tools to do what he did suggested what he had back then was more primitive, so then it was near impossible for people to even dream of such things, and yet Ramanujan did! The question that still bothers us today is, HOW?!?!?!

Anyway, back to me. In Year 4, which is where I am right now, I’m still learning a lot of new things everyday. Several of the interesting concepts in Math come from various marvelous aspects – for instance, using local properties to deduce global properties, using continuous objects to count something discrete, realizing that there are always two (or more) sides to a definition than the single tunnel-vision that we tend to have while doing Math, etc. My supervisor told me that Math is a combination of intuition and rigor – you think something is what it is, and then you verify it rigorously. However, I’ve realized that my attempts to view things in a more intuitive approach too much has some downsides – I realized that my rigor has declined – I’m always forgetting to fill up gaps, or knowing why things can be true but not knowing how to prove them. These things sometimes made me feel very inadequate, but I trust what my supervisor has told me – a combination of intuition and rigor – find out why things are what they are, and then find a way to verify them. That advice has helped me a lot, not only in the exams, but also in my learning.

Another thing that I’ve learned is that while theory is certainly important, we cannot ignore computations. Often times we get too involved in theory that we lose the general feel of things. For instance, in Galois theory, we may study too much about the theory of extensions and its results, but when asked to come up with examples, or think about why things are so, it is difficult. It is then you realize that you haven’t really learned much  – Furthermore, it is interesting to think that Galois, at that time, did not have any formal algebraic treatments as we do today, so he definitely saw things in a different light. Therefore it may be beneficial for us to try to see things in his setting – i.e. remove the abstract nonsense and try to see what he saw at that time.

Still, at the end of the day, I believe that a balance has to be struck in math – between intuition and rigor, theory and computation, and so many more. Looking ahead, who knows what’s in store for me? I can only wait and see, and when things come, try to catch them all at once and internalize everything.

~Falcon, OUT!

Categories: Buzzes in my head, Math

Imagine

November 10, 2010 Leave a comment

Imagine one day you wake up. It is raining. You make your way to the bathroom, brush your teeth, take a shower, get changed and get ready to go out. You peer through your window. It’s still raining. You thought you’d wait a few more minutes for the rain to get lighter and hopefully subside. Anyway it is just a drizzle. It must’ve been raining the whole morning, and now at its final stage.

As you sit on the chair in your room, you wonder how long it will take till the rain dies down. Every 10 seconds you take a look out the window, hopefully looking for signs that the rain is no more. Yet it is still there. It’s still there! Why wouldn’t it go away?!?! Ahh this is irritating! You feel as if you’re a young little bird trying to break out of your egg and enter the world, but humans have glued your shell together so that you can’t get out, and there’s nothing you can do about it. Argh!!

Finally you decided it’s not gonna subside anytime soon. You then grab your umbrella and walk out hoping it stops raining as you walk. To your disgust, the more you walk, the heavier the rain gets. You finally reach your destination, the tip of your pants damp, your shoes wet, your drenched socks chilling your feet more than you would like it to. What’s worse, when you actually arrive at your destination, the rain stops. AHHHHHHHHHHHHHHH!!!! You feel like taking your umbrella, pretending it is a lightsaber and slashing everything in your path!

If you knew the rain would have stopped at THAT time, you’d probably wait. Heh, you’d probably have been able to get more sleep! But the fact remains that we are not God. We can’t see the future. Therefore we take risks. But we need to wait for the right time and place to do it. But when is the right time and place? When is it best to make decisions? When is it best to execute them?

Sometimes we wished someone had the answer to everything. Then, when we are faced with a problem, all we do is confide in that person and seek his advice. He’ll have the answer to everything. He’ll tell you what you need to do, and then you just do what he says and you’ll be alright.

But, gravity pulls you back down to reality and you realize that if there was such a thing, then life would be ever so simple. But it is NOT!

Tell ya what though. You may think there isn’t such a person who has the answer to everything. Well, actually… there is.

~FALCON, Out!!

Finals

November 7, 2010 Leave a comment

So, as you may (or may not) know, I normally do an interview before the finals begin, describing my thoughts and feelings ahead of the exams. This time, though, I thought I’d do it with a little twist – I’ll pluck out some of the questions from the F1 interviews and translate them to its proper context. Alright, without further ado, let’s SUIT UP and begin!

Module 1: MA3209 Mathematical Analysis III
Do you think you have enough performance in you to challenge the top performers in the class?
Well, I don’t know actually. We try to push as hard as we can, but at the moment it seems like Dougie and co. are way ahead of us. But we still have about three weeks, so we’ll be pushing as hard as we can. We haven’t gotten our midterms yet, so let’s wait and see about that and see where it puts us.

Do you think you will need DNC to screw up in order for you to get the result you want?
Ah, well… Obviously I wouldn’t want to see bad things happen to other people. It is definitely not something I would wish for, but if it happens then it happens. But at the moment I don’t see it happening. They are really really in form right now and I don’t see them slowing down at all. Of course, it will not stop us from pushing and working as hard as we are right now. For me, as long as the exams are not over yet, we still have a shot.

So could the other students in the class become a sort of buffer preventing you from getting a bad grade?
What I’ve come to realize in my three years or so here is that you really cannot underestimate anyone. One moment you think this is easy and you’ll breeze past through it, and the next thing you know you’re starting to sink while others already caught their breaths. I don’t know a lot of them and I don’t know what they’re capable of, but if there’s anything I’ve learned in the past is that no matter how easy it is for you to pass them, at the end of the day, you STILL need to put in effort and get the job done. Just thinking it’ll be done is very naive and it’s just taking things for granted, which is an attitude we cannot afford, especially not now.

Module 2: Complex Analysis II
Does your midterm score feel a little like it puts you in the driving seat as you seek to gain more from the final?

The midterm is one thing and the final is another. We have learned that on a lot of other occasions, sadly, the hard way. But yeah, we got a good midterm score, could have been better but we’re satisfied with it. However we know the finals will be tough, no doubt about that. We don’t really know what’s going to happen now, we’re not that confident about it. All we can do is keep on pushing and hope for the best.

Why were your classmates scoring higher than you in the midterm?
They’re smarter. There’s something about them that they can understand the material and internalize it more quickly than I can, and at the undergrad stage, it’s really critical to be able to do that since there’s so little time to get yourself ready. So I guess I’m slower at picking up new things and I believe they will continue to excel in the final as well.

What are your feelings about getting an A+? Mathematically you are still in it, but is it realistic?
I don’t think it’ll be easy to get an A, let alone an A+! I’d say we’re struggling now, and we really need to get ourselves back on track and really push for this one. We have a good foundation to build on, and if we do it right, the results will come.

Module 3: Galois Theory
Was there one particular reason why your midterm didn’t pan out as you’d expected?

Not really, no. It just happened the way it did. I’m disappointed but I don’t want to dwell on it.

Who will be your biggest threat?
Everyone. Practically everyone. There are only 9 students in the class, last I checked, and so I will have eight other people competing with me to get what we all want. We just need to get ourselves relaxed and don’t think about it too much and just focus on the task at hand.

Which is…?
To work hard and nail the final.

Falcon, OUT~

Categories: Interviews

Tick Tock

November 3, 2010 Leave a comment

There’s so much to do, so little time!

On a different note, the awesomeness of this is legen-wait for it—————-DARY!!!

I especially love this part:

你说的一切 我都相信
你握着我的手看着我的眼睛
我比谁 都确定 你的心

Great stuff. Even greater than the fact that I now have an algorithm to compute the modular equation of an arbitrary degree!! That’s right people! The great pfalcon has done what few people have!!!

KK. That’s it for now.

Falcon, OUT!!

The New Song

November 3, 2010 Leave a comment

Just wrote a new song. Ok, not so much wrote, but tweaked. It’s about two guys named Jeffrey and Wang Buo. Check it! (to the tune of Pinky and the Brain)

Wa. Really don’t feel like doing anything.
Then don’t do lah……..

Jeffrey and Wang Buo, Jeffrey and Wang Buo,
One is the genius, the other’s… also a genius!
Their laboratory guys, their brains have been spliced,
Their Jeffrey. Their Jeffrey and Wang Buo Buo Buo Buo Buo Buo Buo Buo Buo…

Before this day is done, their plan will be unfurled,
When you hear a loud boom, they’ve burned down the whole room,
That’s Jeffrey and Wang Buo, That’s Jeffrey and Wang Buo,
Their daytime campaign, is quite hard to explain,
They haven’t proved their worth, but they’ll burn down all the earth,
Their Jeffrey. Their Jeffrey and Wang Buo Buo Buo Buo Buo Buo Buo Buo DONG!!

Awesomeness rules this song!
Falcon, OUT~