Thinking Aloud

…starts with a single step.

My supervisor has finally given me my first task. For those who die, die must know what I’m talking about:

" Calculate the K-types for the theta lift of U(p,q) to U(r,s)."

Treading excitedly and nervously.  

" Eh so what are you doing for a living?"

" Well, I’m doing a higher degree…"

" Oh, that’s great! What’s your thesis about."

" …"

That was a fictional conversation actually. But one that happens fairly often in real life. I remember giving a reply, the condescending and lame: " It’s rather hard to explain it to a layperson." What was I thinking! The guy was a lawyer, I should have put in more effort to explain what my work is all about.

Ok so here it goes.  

Basically what I’m studying is something called the Weil Representation, or sometimes called the harmonic or oscillator representation. This has connection to quantum mechanics; as this representation comes from the study into the equivalences of the Heisenberg picture (which invovle matrices) and the Schrodinger picture (which involve solutions to a certain partial differential equation).

This representation was discovered (among others) by Andre Weil.  This is a representation of the  double cover (called the metaplectic group, abbreviated M2) of the group Sp(n,R) onto the Hilbert space L^2(R^2n). As you probably can see, this is a fairly "large" object. So large that if G and G’ are two groups sitting inside the M2 satisfying certain relationships, a representation of G may lead to another representation of G’ both contained inside the Weil representation.

This relationship was made explicit by Roger Howe and is now known as the local theta lift. This link will take you to a document written by Zhu Chengbo, a Professor with the Deparment of Mathematics at NUS. It explains what the local theta lift is in semi-technical language only in the second section.

But let me try to test my understanding by putting things into layman’s terms. The Weil representation leads to an action of the group on a dense subset of this representation called the (g,K)-module. The good thing is that this module has an explicit realization as a space of polynomials. Polynomials are simple things to understand and manipulate. It is this utility that Howe exploited.

Remember G and G’? If they satisfy a certain relationship, then we call them a (dual) pair. If a representation of G (call it A) sits in the Weil Representation in a certain way, then we can be sure to find a unique representation of G’ (call it B related to A) which also sits inside the Weil Representation. This association of A to B to called the local theta lift.

My thesis is an investigation of the local theta lift for the dual pair U(p,q) and U(r,s). Well, there’s a mouthful. Something to chew upon over Christmas?

This is an interesting article. Compares and contrasts the thinking process in a fundamentalist mindset and the mindset of a chess player.

I couldn’t resist adding this as well. The mind of a chess player may be just as similiar to the way a mathematician works.

Mathematics is about nuances. Although many people have this perception that mathematics is boring, just about numbers and logic, dry and overtly tedious.

But in fact, those are exaggerations. I’ll never get tired of saying this: Mathematics is beautiful. Its like chess. The rules of logic are the rules of chess: They tell you how to move. Strategy in chess is like the overarching theory. Tactics are comparable to the techniques of argument that we use to prove stuff. Getting from hypothesis to conclusion is like painting. First you sketch, then you ink, then paint and frame and enjoy.

Yes mathematics is complex, as with chess, as it is with life. Sometimes, I can’t solve problems because they are too hard for me, sometimes I get them wrong. But the practise and experience in thinking gives me that extra edge in noticing little things that a normal person would have otherwise passed over. This is nuance, this is a way of looking at life and being able to be precise and yet flexible.

I wish that there were more Malaysians who is willing to look beyond pragmatism and take up this beautiful subject as a life study.  

 

Nah, joking onli lar….

But one thing’s for certain. I’m sure I learnt more real analysis in one semester compared to two years worth of advanced calculus to (that nightmare!) undergraduate real analysis.  

A/Prof Chua pushes us hard. But I can’t disagree with his method. Zygmund and Wheeden’s excercises have made me appreciate the subtleties that I wouldn’t have noticed otherwise.

Now, the question is whether I should do Analysis II. We’ll be covering functional analysis. If I do, it’s for certain that I won’t be taking any other module.  

The cardinal truth of life:

In mathematics, the most straightforward solution is usually the correct one…yet, the most straightforward questions are deceptively easy.

In life, the truth is obvious…it just that we have a hard time swallowing it.

I think Grissom of CSI fame quoted something to that effect.  

Haha, I survived teaching a class on the Fourier series.

When I know next to nothing about it.

Haha indeed…

After handing up the third assignment, I begin to dread the next one. 20 questions of untold difficulty waits.

Okay, maybe it’s a good thing since my computer is down and I might as well spend the time doing analysis. But here’s the sucky part, it comes at the cost of neglecting algebraic topology and my other research reading. (Thank God for the foresight of dropping logic)

Anyways, have been rather tired lately, hopefully the semester break will provide a recharge so neccesarily needed.

Back to school tomorrow.

The only difference is that I don’t get the luxury of waking up late and going back home when I feel like it. Not that’s there’s a card we need to punch in or something like that, but there are things to learn and stuff to do.

Heck, one of my lecturers has actually assigned homework liao.

Still, my stipend is conditional on my results. Slacking is definately no-no now. Getting bad grades is NOT an option anymore.  

But for the love of numbers…its well worth the hardwork and stress.  

Spent most of the day thinking about lunch and dinner and one-parameter groups. The Lie group stuff is slowly becoming clear, but its only the beginning…

The still so much more to go…exponential maps, integration, volume forms, fibration, covering…wahlau!

The good news is that I don’t have to start on my thesis straight away, since I’m taking a rather relaxed route here. Most of my cohort mates are doing a one year master’s programme. I’m stretching out the duration as long as I can.

So then, I’m planning to cook something spetacular tomorrow, if it turns out well I might even post the picture of the dish up here… 

 

 

Time to hold my breath.

The deed is done, now everything hinges on God’s almighty hand. If He wills, then nothing can stop it from passing.

What is faith?

I guess faith is like love. It is beyond definition, but it is always recognizable.

For a long while I thought I had faith, that I dared to dream. But this semester I discovered that all these past three years I never went beyond myself; be it in studies, hall activities, student ministry. I played safe. I thought I was confident, but it was an illusion. A safety net constructed so that I will not have to risk failure.

But this time it is different. Failure is defined in black and white terms. There is no hiding, no way to justify, or outreason the consequences.

Do I have faith? I have learnt that I have none. True faith trusts, and one truly trusts when one is not in control.

There are things which I have no control over, and it is in these things that faith is made real.

And I know I can trust Him, because He loves me.

Cos I have to take my Logic exam tomorrow.

Bleah….

I spent the whole day doing only ONE!!! miserable number theory question. Not because I was lazy, but simply because I was blind.

Sobs, I’m so going to die on wednesday (next week!)…

and I haven’t even revised Logic yet.

Exams start today. All the best to the set of NUS students intersecting with the set of the readers of my blog.

Whew, that was verbose…

And how did I spend the whole day?

Reading this…

which has no relation at all to my examinable subjects.

I’m committing intellectual suicide here.

(for those who know, this little excerpt refers to the sheafification construct of a presheaf. We just look at the continuous section of a stalk which is the direct limit of of presheaves on a system of open neighbourhoods about a point x in a topological space X.)

Math’s galore…I’m mad already…hehe…

Who ever knew algebraic geometry is so botanical…sheaves, stalks, sections :S

Since I’m speechless, I only want to share these few verses.

You have turned for me my mourning into dancing;
You have loosed my sackcloth and girded me with gladness,
12That my soul may sing praise to You and not be silent
O LORD my God, I will give thanks to You forever.

For His anger is but for a moment,
His favor is for a lifetime;
Weeping may last for the night,
But a shout of joy comes in the morning.

For I know the plans I have for you,” declares the LORD, “plans to prosper you and not to harm you, plans to give you hope and a future.

If you happen to know more verses in the same vein, leave a message on the tagboard.

At this moment in time, I just look back and whisper, ” Time really flies.”

But then again, isn’t it true that yesterday seemed like a dream from a century ago?

It was just last semester that I first stepped into the world of ‘real’ (being relative) mathematics, but now that dream is going to be cut out from me?

It is really in moments like this that my capacity to trust Him is tested to the limit.

If only the Maths Department would hurry up and tell the results of my application.

Tunggu, tunggu, tunggu…leceh betul lah.

Haih….

Two phycists, Mr. Brown and Mr. White and a logician Mr. Orange are wearing hats. They know that the hats are either white or black but not all are white.

Mr. Brown can see Mr. White and Mr. Orange hats.
Mr. White can see Mr. Brown and Mr. Orange hats.
Mr. Orange is blind. (poor him!)

Each are asked whether they know the colour of their own hats. And they each answer

Mr. Brown: “No.”
Mr. White:” No.”
Mr. Orange: ” Yes!”

So what is the colour of Mr. Orange hat, and how on earth did he know!!

**********
Let me assure you that this has a valid answer, and I know it.

A list of the modules I’ll be taking this semester.

Graduate algebra,
Intro to analytic number theory,
Mathematical logic.

So now I’m an algebraist, number theorist and logician. So tell me, what is truth?

Swan beauty, majestically floating down the stream.

Ain’t it pretty.

Ok that’s about swans, but now to serious stuff.

1) The holy grail of contemporary pure mathematics is the so called Langlands programme. Much exciting development is happening here.

2) This is a field comprising number theory, automorphic forms, class field theory, algebraic geometry and the super power mathematics that undergraduates only dream about understanding.

3) This is the field of research that I hope to immerse myself into.

4) Therefore, I need to work hard.

5) And hopefully, God opens doors…

I’m treating God like Santa Claus now, How low can I get….

It was a moment to remember. The second last hurdle has been crossed. There’s left only to show the existence of a group G = HK such that H and K are both not normal (in G) but yet each of them solvable. G should not be solvable.

I can only start by considering nonabelian groups, yukk!!

This is how doing group theory for one whole week feels like:

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

I was browing through friendster when I came across a photo of Kayjin with Dr. Tan (my honours sup) taken during his convocation.

Feeling super guilty right now.

*Hangs head in ultimate shame*

Spend the whole day thinking abour Steiner’s Porism. It really felt a little cheating it proving it using complex analysis, somehow the proof (which I think is correct) seems like using a chainsaw to cut sticks.

But I’ve proved it. And I hope the argument holds water. It better…after dying in the midterm, this better be good!

If anything, I really wish sometimes that life is like geometry. Nice and straight, curving at right places and heading of to infinity. What’s more, geometry is beautiful.

That’s just wishful thinking. Unlike mathematics, real life decisions are not cut and dried. One cannot see all ends. Geometry is like Michaelangelo, Rembrant and Picasso. Real life is like Pollock and Koonig.

Sorry Faith, yet another mathematical post.

Category Theory (that’s what I’m learning now) is really sticks and dry bones. Its interesting, but difficult to see why its so important in modern mathematics.

Here’s something to go with your coffee and eggs in the morning.

Do you know the answer to this?

1) If you have 3 odd prime numbers, is it always true that there is at least one of the three that divides the product of the other two leaving remainder one. ( Let p,q,r be three distinct odd primes. Is it true that at least one of these cases happens? qr = 1 mod p, pq = 1 mod r or pr = 1 mod q.)

I think it’s true, but can’t prove it.

My parents are teachers. But their passion and gift sort of didn’t rub on to me. I absolutely have a phobia of stepping into a class of Form 5 students and giving a lesson of trigonometry. Can die wan!

I will have to give a short presentation on the analytic continuation of the Reimann Zeta function sometime next week or this thursday. The thing is, will my presentation be well recieved? Will my fellow coursemates be able to take away something from it?

I guess, for any teacher (or lecturer) the art and trick is to give the essentials without sacrificing the depth. Some teachers manage to do that. I still can remember how Miss Moey thaught us Bible Knowledge in form 5 and 4. Inspirational!

I really wonder how is it that a particular human being is able to connect so well with another and yet some are not able to. Charisma indeed is one of the intangibles of human existence. How is it that business consultants and trainers are able to whip up a storm during seminars? How is it that (some) pastors are able to preach effectively? All these are a mystery.

The wonder of a teacher is how he/she is able to be transparent; letting the student probe his/her mind without being confused.

As it is, the essentials without sacrificing the depth. The great and noble profession, teaching. No wonder experienced teachers know that they must love thier students. It is the nessecary condition to be able to even establish the basic rapport that allows learning to take place.

Yep, the season for Intro Talks have begun. Mine’s tomorrow and there’s one starting today. This whole week and the next will see the Colloqium Room A fill in and out with honours students presenting their power theories.

Wish us all the best.

I have one prayer request from all those who read my blog. Just pray that I’ll be able to make the fullest use of my time back home. That’s all I need really.

I really need to be more organized. My desk is a mess and I feel so unkempt. Hopefully at home I can use the “get-away” to refresh myself. I need it. Feel like burning out liao.

I think I’ve finally got it (Prof Berrick’s super vague tutorial) after a long night of schizoprenic muttering. It’s all in the interpretation of what a matrix means. Oh well, thank God for that.

And to answer Marcus query: My intro talk is tentatively on the 27th Tuesday, September from 10-1030 am. All are welcome (not just maths majors). Keep a lookout for emails from the Maths department.

Kay Jin didn’t show up for the tutorial discussion today. GRRR! Anyway, I thought could get him to help out in the parts that I didn’t understand. Hieu and Venku weren’t much help. Sigh…this is sooo distressing.

Now how?

Slog through tonight to make sure that at least can answer some of qn two lar…I mean there has to be something that I am not looking at. Something that I’m missing.

I’ve totally no interest in qn three, though it might turn out to be the easier of the four. Qn four looks interesting and I have some inkling of how to solve it but that’s for the last.

Bummer….really bummer.

Really, not only isn’t there a spoon, there isn’t even a holiday.

Why?

Group Theory Homework 4 is due on the 24th, I have a test on the 23rd and 29th, I have to prepare for my Honours Introductory Talk and prepare for a small presentation on the Gamma Function.

Sweet right.

Come to think of it, I actually only have 4 days at home only. That’s not much…oooo how I love NUS.

Prof Berrick says,” The nice thing about teaching graduate students is that you can be so cruel.” and proceeds to write on the board after presenting a theorem: Straightforward.

The presentation on projective modules was minimalistic so to speak, one could almost say zilch although there was something going happening on the board.

Was taking graduate algebra a mistake? The midterms on the 23rd and God knows what’s going to happen.