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## The moment it clicks

One of the most beautiful aspects of doing math is the flash of insight that lets you out of a mental block.

This may appear surprising to those not in the field but the fact is that research mathematicians fumble around looking for the light switch in a dark room for much of their waking hours. Things usually don’t work out. Being stuck on a problem is the default state.

That’s what makes those moments precious. Suddenly, you see it. A germ of an idea appears in your head, something clicks and you realize — this will work.

My roommate has a wonderful book on photography whose title is the subject of this post. The book has amazing photographs but I also find the title excellent. It is all about the moment, in art and in math.

## The pursuit of science and an old email

I stumbled upon an old email today. It was written by me in December 2001. That’s very long ago, isn’t it?

***

Suppose you ask me today … (no, this is not part of the email)

So, what’s worth pursuing?

Well, scientific knowledge is worth pursuing.

What about happiness, money, comfort, security?

Sure, all those too.

But what’s most worth pursuing?

Dude, it’s a matter of choice. Ever heard of freedom?

***

Well, on the other hand you could ask me…

In what way is your worldview substantially different from the person you were seven years ago?

Hmm, that’s hard.  How about greater relativism in my assignment of worth to other people’s goals?

But that’s coming back to choice again.

That’s right.

***

But I am getting ahead of myself.

There’s an institution in India called the Indian Institute of Technology. It’s usually shortened to IIT. When I was in school, it was commonly regarded as something of a holy grail by my peers. As the entrance exam for this place is absurdly competitive, only the best students have a decent chance of getting through. And most of them indeed make this exam their sole focus during their last two years of high school. This whole IIT thing bugged me to no end. To me, the only reasonable target for someone gifted was the pursuit of knowledge. In other words, research.

Now, these smart kids who were single-mindedly preparing for the IIT entrance test had no great love for engineering. Most of them were pursuing the IIT dream either because their parents forced them to or because they believed (correctly) that it would assure them a plush job. An IIT degree was proof to the employer that you were highly intelligent and hard-working — ergo, suitable for any job. It is not uncommon for an IIT graduate in chemical engineering to be wooed by the advertising industry, or a mechanical engineer to take up a job in the financial sector. IIT was your passport to a good life. And in my eyes, it was an abomination.

You see, I was a research fanatic. I simply could not fathom why a gifted young mathematician or someone deeply interested in the physical sciences would choose to go to this place. Everytime I visioned a smart person taking up a managerial position in some firm, something died in me. To me, it was the equivalent of selling your soul.

(True, a minority of students go to IIT and still pursue a research career. This is especially true in fields like CS. However, as far as I was concerned then, it didn’t happen at all.)

***

So, when I discovered that a very good friend — who was also very talented in mathematics — was preparing for the IIT entrance examination, I wrote her a long email attempting to dissuade her. The magic of technology meant that a copy was preserved and I stumbled upon it earlier today. It is an interesting document, not because it has any new ideas or is particularly well-written but because it showcases the passion and reverence I had for the idea that everyone with enough talent should do scientific research.

With age I have mellowed. Some of the prejudices that the email displays have little in common with the person I am today. Yet in some ways, that email is also very me, and it mildly saddens me that I would never write such a thing today.

For the purpose of anonymity, let’s call my friend M. It’s the first letter of the name (not her real one) I used to call her by. I also loved her, but that is a different story.

***

Dec 27, 2001

Well, about IIT, its a very good engineering institute. An excellent place for meeting intelligent people and studying really hard and learning methods to solve horrendous differential equations. And after you pass out, unless you are very unlucky you will get a much-coveted white-collar job in a slick office where you will spend ten hrs a day signing documents and contracts and tenders, attending meetings and handling lots of business files. And at the end of each month, you will get a nice fat paycheck…

M, I am advising you to study maths not because IIT is bad. But because you are good.

I believe that if a person is really good at a particular subject and has a deep interest in it, then he or she should pursue higher studies in that subject.The best should do research. Engineering (or perhaps, in view of the kind of job people actually do after passing out of an engineering institute, I should say ‘pseudo engineering’) can be left for the others.

After all if a student loves a particular subject and has a real talent in it, it is only logical that he should aim to contribute to it!

Of course I am all too aware that the vast majority of talented students join IIT. For two years thousands of bright students prepare crazily for the IIT-Jee, join coaching classes to get into coaching classes which coach them for the iit-jee, the holy-grail of all examinations. IIT for them is the ultimate destination. Indeed this unbelievable IIT-madness is an amazing sociological phenomenon-probably unprecedented in the history of any country.

Its also a vicious circle of the most heinous kind. Somehow, everyone seems to think that you have to aim for IIT– to even think of anything else is either a joke or an outrage. Its like the rats of Hamlin, all of them swarming into the ocean at the tune of the piper without knowing why.

Brain-drain is a term commonly used to refer to the migration of the best Indian brains to foreign countries. But here we witness the drain of virtually all the best brains of the country from mathematics,  physics (and other subjects) to air-conditioned offices where they do semi-clerical work. Like the rats of Hamlin, it is a phenomenon so absurd as to be almost laughable. Except that its not possible to laugh at something so serious.

And that’s a shame. Of all the shames plaguing the Indian education system, it is the worst.

M, I know that you love maths. And I know that you are good in it. Would you really like the kind of job that you will be probably be doing after passing out from IIT?

Of course an IIT-pass out is paid more than a mathematician. But trust me, as a mathematician you will be paid enough to lead a comfortable life. And above all, you will be doing something you love. You will be doing mathematics — making contributions of your own to the subject and teaching the subject to others. You will get plenty of leisure time. And if you make a truly significant contribution to mathematics, the kind of recognition you will get will be beyond anything you can expect to get from doing the kind of work an IIT pass-out does.

Immortality may be a silly idea, but a mathematician has the best chance of achieving it.

***

M ended up going to MIT for her undergrad. She is currently a pursuing a PhD in theoretical computer science.

## Late night incoherent rambling

The posts on this blog are mostly socio-political commentary from a libertarian perspective which might lead regular readers (if any exist) to falsely conclude…

It is almost 2 am now. Having spent the last two hours reading old posts on “Overcoming Bias” and other excellent blogs (some of which, like this one, I just discovered), I am fatigued. Yes, those things are important. Yet, if I were to make an informed guess, I’d predict that my degree of passion for political matters will one day become similar to those of EY who writes:

I started my career as a libertarian, and gradually became less political as I realized that (a) my opinions would end up making no difference to policy and (b) I had other fish to fry.

…that my biggest fish are reflected by what I blog about. But people are mortal, all political theories imperfect, all systems of government temporary. Isn’t is obvious then that math is of far greater value?

## Update on Riemann hypothesis

A couple of days ago, I wrote about Xian-Jin Li’s claimed proof of the Riemann hypothesis. Here are matters as they stand now:

Li’s preprint on the arXiv is now up to version 4. The initial problem identified by Terry Tao is now fixed; however a more serious problem pointed out by Alain Connes remains. Basically, Li’s paper appears to adelically integrate a function supported on the ideles, which cannot give anything useful because the ideles have measure zero in the adeles.

Terry Tao explains here why an approach such as Li’s, which basically only uses the multiplicative structure of the adeles, cannot work. Money quote:

The argument only uses the multiplicative structure of $k^*$, but not the additive structure of k. (For instance, the fact that k is a cocompact discrete additive subgroup of A is not used.) Because of this, the arguments would still hold if we simply deleted a finite number of finite places v from the adeles (and from $k^*$). If the arguments worked, this would mean that the Weil-Bombieri positivity criterion (Theorem 3.2 in the paper) would continue to hold even after deleting an arbitrary number of places. But I am pretty sure one can cook up a function g which (assuming RH) fails this massively stronger positivity property (basically, one needs to take g to be a well chosen slowly varying function with broad support, so that the Mellin transforms at Riemann zeroes, as well as the pole at 1 and the place at infinity, are negligible but which gives a bad contribution to a single large prime (and many good contributions to other primes which we delete).)

Li would probably have been better off showing his work to some experts before releasing it on the arXiv. As any mathematician would testify, it can sometimes be surprisingly hard to realise your paper contains a mathematical error, especially if you are confident of its correctness.

Update(July 5): Li has now withdrawn his paper from the arXiv.

## Another attempted proof of the Riemann hypothesis

They keep cranking them out, don’t they?

Of course, there is the minutest possibility that this time its for real, but just going by history, I am prepared to bet all my savings against it.

Update: I apologize if my post seems to suggest that Xian-Jin Li, the author of the purported proof, is a crank. In fact, he is a competent mathematician who has done good work in the past. Nonetheless, I think that the chances of this proof being correct are extremely low; in fact Terry Tao claims to have already found a mistake.

Update 2: Xian-Jin Li has posted a new version (actually two three new versions!) of his preprint on the Arxiv. Most pertinently, the definition of the function h on page 20 has changed; so perhaps this addresses Tao’s objection above.
The reason I will be very surprised if this proof turns out to be correct is that it involves mostly functional analysis on the adeles. It has been generally believed that such techniques are not sufficient to prove Riemann. It would be a stunning achievement if Riemann is solved using only such elementary tools; will be following this news closely over the next few days.

## Number theory for the mathematically curious

Most people, on coming to know that I do research in pure math, respond with a nod or a wide-eyed, “Ohh, that must be so hard!” Occasionally however, someone goes further and asks me what my research is really about. And then, I am usually in a fix.

How do I respond? There’s no way to explain that I study special values of L-functions for automorphic forms to a person who is not already familiar with all these words. So I usually take refuge in generalities like “Prime numbers”. Sometimes when I am in the mood, I explain to them what Fermat’s last theorem says (if I am lucky they already know this) and add that I work with methods ‘related to’ how Fermat was proven.

It was therefore a source of great joy to me to read Barry Mazur’s excellent article in the Bulletin on error-terms in number theory and the Sato-Tate conjecture. While the article isn’t quite about what I do research on, it comes fairly close. More importantly, it is engrossing, beautifully written, mathematically solid and accessible to anyone who knows some college-level mathematics and statistics. Perhaps, I should start carrying a copy of it in my pocket for exigencies like described above.

## Free to create

“The essence of mathematics lies in its freedom.”

George Cantor.

## The critic and the creator

“It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.”

G H Hardy, the opening lines of A Mathematician’s apology.

## Lang-Bang!

So how is it listening to Serge LANG?

For starters, awesome! On second thoughts, phew!

Well, Lang is a guy who actually jumps around while lecturing and every five minutes or so, comes up to a member of the audience and asks the poor creature questions!

So Jennifer was sitting there along with the rest of us, when Lang came up to her and started ,”So tell me..”

She, being female, sensed the trouble before the rest of us. “NoNoNoNoNo…”, she cries out.

Lang, is unmoved. He completes his question, tells Caltech(and the world in general) that she would have known the answer if she had only read Analysis from Lang’s book, and resumes his talk.

There were around 15 of us, including several professors, in a small Caltech classroom listening to Lang talk on “The Heat Kernel and associated zeta functions”. It was fairly elementary, and Lang is a nice speaker, but I am lucky to be one of the few who escaped the Lang sabre.

But the talk was great fun. The funniest exchange is as follows.

Lang(bellows) : If the writers of undergraduate textbooks in analysis weren’t incompetent idiots, all of you would have already known about the big bang of mathematics, the heat kernel!

Member of the audience, a professor: So is it there in your analysis book?

Lang: Well, it was in the appendix in the old edition, but, in the current edition, it is there in the text.(Smiles mischievously and walks up to a professor) Ok Sir, tell me, by Aristotelian logic, what can you therefore conclude about me?

Professor squirms and refuses to say anything.

Lang: That is not good! He wants to take the fifth amendment! Ok, you(points to a grad student), tell me, what can you conclude?

Student: You are no longer an incompetent idiot!

LOL!