What is the smallest uninteresting number?

You can try to find the answer on Wikipedia. The smallest number without its own wikipedia entry is 224 (as of today). Despite it being the sum of four consecutive cubes (224 = 8+27+64+125), it only manages to share space with the article on the number 220, appearing in the "221-229" subsection.

Surely that's a lousy criterion! Sooner or later, some bored guy behind a computer is going to write up an article on the number 224, following which, 225 would become the smallest uninteresting number. That's despite it being more interesting than 224... after all it is the sum of five consecutive cubes.

It might be better to ask an actual mathematician. G. H. Hardy, the British mathematician who wrote "A Mathematician's Apology", was visiting his advisee, the great Indian mathematician Ramanujan, at a hospital. Being the typical awkward conversationalist, Hardy began by remarking on the dullness of the license plate number of the taxi he was in -- 1729 --- and was hoping that it would not turn out to be an unfavourable omen. To which Ramanujan replied, "No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways".

I learnt of this lovely anecdote while watching QI, which is short for "Quite Interesting", which is a most interesting show on the BBC where Stephen Fry and four blokes would converse about obscure trivia --- fascinating, funny and most definitely repulsive to the pragmatist. 

According to QI, the "correct answer" is 12407. This sleep-inducer holds the distinction of being the smallest number absent from all 200000 sequences in the Online Encyclopaedia of Integer Sequences. That is to say, it's not interesting enough to appear in any number sequence that mathematicians might be interested in.

But surely, that is interesting? To be the smallest uninteresting number is probably one of the most interesting attribute any number can have! And so we have a paradox. If there were a smallest uninteresting number, then it would in fact be interesting by the mere fact that it is so small and so uninteresting. Therefore, there can't be a smallest uninteresting number.

And so what have we gotten ourselves?

For all the math-haters out there, we have a proof that all numbers are interesting.

(I am of course, referring to natural numbers 1,2,3,... As always, the real numbers are a different beast altogether, and one might be able to argue that there are uninteresting real numbers.)

Oxbow tie

As demanded, here are a few pictures of myself in a silly costume.

Spot the Potter

Front view indoors --- with a pipe organ to match

Front view outdoors

Side view outdoors --- with a lady to match

Got caught in a ridiculous hat

Lunch

Something earth-shattering happened today. I made my own lunch...

... and I took a picture...

E=mc^2

This post was initially intended to be a one-liner:

Do we really need another atomic bomb to appreciate the relativity of space and time?

Everyone knows that the most famous equation in physics is E=mc^2. Most people know that it is related to nuclear-ish things and the inexplicable thing called "relativity". Some people know that electromagnetism led to Einstein's special relativity, when he pondered what light would look like, if he was chasing after it. I discovered, to my greatest horror, that I don't quite know where E=mc^2 comes from!

One might find it amusing that this embarrassing realization is keeping me up at this unearthly hour (it's 4.30am now); but just imagine how Einstein managed catch any sleep at all after deducing E=mc^2 and its corollary -- the atomic bomb. After all, humans have been known to go to war, pre-emptively, over nuclear weapons.

Anyway, I suspect that he might have lost even more sleep upon realizing that "...henceforth space by itself, and time by itself, are doomed to fade away into mere shadows..." 


But, really, why is E=mc^2 ?!


I spent a long time on google trying to find the answer, but mostly found unsatisfactory circular arguments. I also discovered that the meaning of E=mc^2 is not at all clear. For example, it is somewhat of a myth that mass can be converted into energy --- indeed the popular explanation for the mysterious "source" of atomic or nuclear energy. Another popular saying is that mass and energy are two forms of the "same thing". What does that even mean anyway?! Mass and energy are the same thing......!


Perhaps one should first recognize the distinction between matter and mass. These are more or less the same thing for Newton: matter has mass, stuff that has mass is matter. But in Einstein's relativity, mass, and even "rest mass" does not really behave like matter. (That's not to say that it is clear what "matter" is, in the first place; suffice to say, matter is "stuff" and mass/energy is some property of "stuff".) The rest mass of a system of two objects need not be equal to the sum of the individual rest masses! Anyway, the point is that it is more accurate to say that binding energy, rather than mass, is converted to radiant energy in an atomic bomb. 


The upshot of all this babble is rather shocking. Assuming Einstein's own postulate about relativity (i.e. the speed of light c is a constant), E=mc^2 is nothing more than a definition of (relativistic) mass in terms of the energy E. One then asks what energy is, and this is truly a deep question. My own understanding is that it appears, via Noether's theorem, as a conserved quantity corresponding to the symmetry of physics under time translations.


So, is the most famous equation in Physics nothing more than a smoke-bomb?

I can't give a definitive answer, since I'm rather confused myself, and also quite groggy. Einstein himself derived the formula without recourse to his relativity. He simply imagined a box with a light and realized that mass (understood as resistance to motion) had to be equivalent to energy (multiplied by c^2)! In this context, the speed of light c is not assumed to be a constant, so that E=mc^2 is not sufficient for special relativity. This is also the reason why my one-liner didn't make complete sense when I thought a bit harder, leading to a much longer post than I had originally planned.

Relativity is a bitch.

Locating Kaaba in Mecca

This Ramadan, I am reminded of a funny incident when I first saw the green arrow in a hotel room. I was told that it indicated the direction (Qibla) towards the Kaaba in Mecca.

In the Beginning, determining this direction wasn't much of a problem. Presumably, Muslims were located on a local patch on the Earth's surface, which more or less resembles a flat piece of paper. They probably developed some sophisticated method to determine the Qibla using the stars. Later on, compasses might have been used.

But the surface of the Earth is not flat; it cannot even be deformed nicely onto a flat map. As one moves further away from Mecca, the ambiguity in determining the Qibla becomes more and more severe. Given a paper map, the quick solution would be to draw a straight line from your location to Mecca. That certainly looks like the shortest path.

If I had a pilot friend, he might convulse at this suggestion, pointing out vehemently that a straight line on a flat map deforms to a "rhumb line" on a globe. But these are not lines of shortest distance on the Earth's surface! One should instead face the direction "as the crow (or pilot...) flies", which traces out a Great Circle. This is a world of difference --- in North America, the former prescription gives Southeast as the correct direction, while the latter says Northeast is correct.

If I had an astronaut friend, he would scoff at these worldly notions. Being a privileged being whose location, unlike ordinary people, is not limited to the surface of the Earth; he has no need for complicated concepts like "geodesics" or "great circles". However, the question of determining the Qibla from space seems complicated enough that conference of 150 Islamic scientists and scholars had convened to settle the issue.

There is an Atoll called Tematangi, which is roughly the antipode of Mecca. The people there are lucky, because every direction seems roughly correct! It's a bit like standing on the North Pole, and trying to decide which direction the South Pole is. But if one is very particular about selecting the direction giving precisely the shortest distance to Mecca, then the "correct" Qibla would vary wildly from point to point. In fact, one descends into a singularity of sorts at the exact antipode of Mecca...


This is where naive thinking might actually serve us better. Certainly the shortest path from Temantangi to Mecca is a straight line through the centre of the earth. So the Qibla, dare I say, is "straight downwards".

Deal or No Deal?

I recently asked some friends a rather lame question (due to Brian Greene):

Suppose you are on Deal or No Deal, and you have just won a prize consisting of an infinite collection of envelopes:
The first envelope contains $1, the second envelope contains $2, the third envelope contains $3,...., the Nth envelope contains $N, .... (ad infinitum). Now, the host offers to double the amount of money in each envelope. Deal or No Deal?

Wiseguys who have taken some calculus or analysis will quickly point out that both offers are worth Infinity dollars, so there's no difference in the two offers.

Guys who realize that they only live for 80 years, or have only storage space for 2,000,000 envelopes, will shout "Deal!", and then grab as many envelopes as they can, whose contents have now been doubled by the host.

Immortal types will realize that the original offer already contains the doubled-up offer, plus other envelopes with odd numbers of dollars. Thus, after declaring "No Deal!", they will proceed to pocket the proceedings from the envelopes until the End of Time.

Mathematicians and string theorists, after assessing their options using various arcane tricks, will depart with "No thanks, keep your money Sir!"
It's not because they are weirdos who couldn't give two hoots about becoming insanely rich. More probably, they are too poor to accept either offer...

This is because 1+2+3+4+.... is really equal to

-1/12.

(Terry Tao says so! See his Eqn (5), or alternatively, Wikipedia.) 

A Cynic --- to be or not to be?

"The more the universe seems comprehensible, the more it seems pointless" --- Weinberg himself.

An amusing SMS exchange with a long lost friend went along the lines of:

Him: Must there be space and time in order to do Physics?
Me: Maybe not. Just their properties would suffice. The points are redundant.
Him: Not even just pointless, but topologyless!
Me: (tentatively: I'm against pointillisme in Physics.)

And I proceeded to fix a place and time to discuss this very issue... what a humbug!

Cam-Cam!

My crib, made of real brick.

A couple of texts, mainly on maths.

My garden, pail's there by accident.

Half of my cleaned-out room, featureless.

The Mathematical Bridge. (Really.)

The Mathematical Tree. (Not really.)

Lastly, me.

The state of my State

"Democracy is simply a system of inter-generational bribery."

Just about sums up the recent elections.

Moldovan paradise

A real life fairy on a unicycle, a standing drummer, and the next fashion craze. Hold on to your hats!

The Big Lie

"You know politics, I know party chicks!"

If you are going to lie, you may as well lie big.

Preferably have it exposed only after you die.
I wonder if developments in the world have outpaced institutions that we have grown up with and become accustomed to. Huge concepts burdened upon us at birth.

Gold today is no longer related to the normal economic cycle of supply and demand, jewelry, Indian wedding seasons, rain in the Middle East. All those things are passé, forget about them. Gold is driven today by one overriding and I am afraid, at least in my opinion, an irresistible and irreversible trend. A fundamental, global and growing insecurity… A fundamental, global and growing lack of confidence of the world in everything they were brought up to believe. Institutions, insurance companies, banks, issuers of mortgages, ratings agencies, equities, sovereign debt, Federal Reserve Banks. Portugal and Iceland. Greece and Spain. Currencies. What is left? What is left?

Obviously not written by me. But substitute "Gold" with just about anything you are familiar with, and it still stacks up --- love, careerism, democracy, capitalism, socialism, money, nationhood, science, religion, Facebook.......(and theoretical physics of course).

One of the first grand concept that I abandoned was religion. I probably didn't think really hard about it.
This would be followed by money. What is money? How does it work? (well.. economics in short). Why do I want it?
Then there was nationhood, which got killed very quickly by National Service. Will the concept of a "nation" persist for much longer?
Currently under intense scrutiny, is Science itself. Is there an objective reality? One universe, or many?

The Big Lie that is unraveling itself, is that the United States has a free market capitalist economy. Got found out, so to speak.
Scientology is another one that is evaporating under Wikileaks.
The entire 9/11 - Osama - war in the Middle East business might be another lie, or a half-truth.
The Nazis purportedly carried out a Big Lie, supposedly killing millions of Jews and effecting a brutal fascist state. Some say this is itself a Big Lie.
Man landed on the Moon in 1969 (purportedly). Unfortunately, we've been looking back ever since.
A bunch of clever physicists devised a very nice model of the elementary particles, so beautiful and profound... except no one else could appreciate it. Missing a certain "Higgs boson" (promoted as the Holy Grail of physics, and the "God particle", and having the intellectual upper hand, they convinced politicians to splurge 7.5 billion euros on the monstrosity called the Large Hadron Collider. Will this turn out to be a Big Lie? A philosophical question -- does the idea of a fundamental particle even make sense?
The Human Genome Project promised to tell us why we are the way that we are. You know, with the central dogma, genetics and all that. Of course that wouldn't have gotten it much funding; it also promised huge advancements in medicine. Turns out, it might also be a Big Lie. (Disclaimer: this is my layman perspective).

Governments have been known to lie..... big. This elections, little bits of filth have been painstakingly dug up and hung for everyone to see. The fear is that the rot runs far, far deeper. We'll see. At least they can be exposed. But if the Universe itself was a Big Lie, could we ever find that out?

Will you berate me for asking David Hume's innocuous question: Will the Sun rise on the 8th of May?

A mathematician's apology

Disclaimer: I stole the title 'A Mathematician's Apology' from the exposition of G. H. Hardy. If you have the time, do read it. It can either be a 50-page rant by a mathematician, or a 50-page journey into the mind of a working mathematician (which has been unjustly presented in the popular culture).

Actually I am lying. Why does the article have to read like one or the other? I had merely constructed a false dilemma. The article could just as well have been as thrilling as a murder story. More likely, you will not even have trawled through the first paragraph, and therefore, do not have any opinion of the piece whatsoever. Or do you?

Recently, I had the pleasure of carrying out numerous discussions with old friends, who have all gone on different paths in their lives, and formed strong opinions about the world, influenced by their respective fields. The general consensus is that logical thinking and argumentation is the way to go. It must have come as a surprise, when I, of all people, expressed doubt about the infallibility of logic. After all, isn't the job of a mathematician or physicist to form, discover and analyse logical statements about the mathematical or physical world?

As I see it, most professionals, and even scientists, hold mathematics and physics in good faith: Physics is a pillar of human knowledge, holding unshakable truths about the universe that we inhabit. The language of physics is mathematics, which is the oracle of last resort, but presumably, always knows right from wrong.

This is where I begin my apology. (1) Physics is not logically consistent, (2) the language of physics is not always mathematics, and (3) mathematics is... not always logical.

(1) The orthodox presentation of modern physics gives two fundamental descriptions of nature: Quantum field theories, and general relativity. Both claim the status of a "fundamental theory", but neither can be derived from the other. This example should shatter the myth that physics is consistent. String theorists will readily tell you otherwise. They say that theirs is a mathematically consistent framework that can incorporate quantum mechanics and gravity --- a theory of everything! Of course, we don't really know how to falsify string theory, and one might argue that it's a matter of taste (or indoctrination!) to accept a physical theory only after intense empirical scrutiny.

I don't preach empiricism, nor do I dismiss it. It is a big question mark to me. Hence, my doubts about the widespread use of "evidence-based" thinking in the real world. Law demands evidence. Medical research is judged on "sample size" and quality of "control subjects". Economics and policy making is often obsessed with analyzing data, numbers, graphs and, God-forbid, "measures" of human desire and emotion. So, here is the confession of a student of physics: I am not entirely sure about the role of experimental evidence in the grand pursuit of the "truth".

(2) Is the language of physics, mathematics? Sometimes. I will not pretend that science is completely honest. In fact, honesty gets thrown out of the window when the really unsettling questions are asked. Consider the following remarks, all made by professors teaching me quantum mechanics:

"There are no theorems in physics."

"Everything in quantum mechanics starts from these axioms: ..."

 "Physics is not mathematics. Mathematics is just a tool."

 "This is what a proof looks like in physics (handwaves and chuckles to himself)..." [meant as a form of dark humour I suppose]

 "Mathematics can be invented... for physics." [I came up with this lousy one myself]

David Hilbert immortalized this dilemma in his "Sixth Problem": Axiomatize all of physics. This is one of the few remaining problems and will earn you both a Nobel prize and a Fields medal, if you are interested in these sort of decorations.

What can we make of the meaning of "truth" and "falsehood", if in our physical laws, we sometimes reject the arbitration of mathematical logic? Do we implicitly concede ultimate judgement to a deity of sorts? Does "judgement" even make any sense in this case?

The problem runs even deeper. Even if physics were to be successfully axiomatized, there are still disturbing problems in the foundations of mathematics. Again, the orthodox treatment (actually I'm not too sure about this --- nowadays, it seems trendy to take a contrarian view), is based on axiomatic set theory. For instance, the numbers that we are used to playing with, have a rigorous (but not necessarily firm) foundation in set-theoretic language. So, discrete mathematics, combinatorics, and computer science have strong foundations in this approach. But even among mathematicians, there are vehement objections to set theory as the de-facto foundation of mathematics.

Set theory has a "constructivist" philosophy --- in order to prove the existence of an object, you have to "construct it". A huge problem arises when one wishes to admit the continuum (implicitly used, by the way, whenever you use calculus). The continuum is so huge, we could never construct it on paper, so we construct it with our mind! The axiom of choice is put in by hand as an axiom of set theory, and voila! We can now say that the real numbers exist. If that doesn't give you a tinge of discomfort, perhaps your new superpower would --- with the axiom of choice, you could chop a pea into little pieces and reassemble them into the Sun! But of course, physics ceases to be based on mathematics here, and elsewhere, where not needed...

(3) Perhaps we simply have not developed physics and mathematics sufficiently, but ultimately, should we not be able to refine all our knowledge into a coherent and logical framework? Godel's incompleteness theorems roughly says that there are inherent limitations to axiomatic systems (I will not butcher them with my inadequate understanding of Godel). Suffice to say, even for the arithmetic which we are all too familiar with, there are undecidable statements. If you are a very logical and observant person, you might have spotted my use of the "Liar's paradox" in the first sentence of the second paragraph. Was I lying? Or was I not? The prudent teacher should stop the inquisitive student at this moment.

Even this may not dull your optimism about human knowledge. Who cares about the undecidable statements anyway? As long as everything is logical... But what... is logic? From here, there are at least two possible paths. The first is that of a mathematician delving in mathematical logic. Theirs is a world of constructivist logic, intuitionistic logic, symbolic logic, fuzzy logic, higher-order logic, quantum logic... etc. Exotica that is far-removed from the everyday logic that we use in practically all human activity, and which we hold so exceedingly dear in our desire to tell fact-from-fiction, and to judge each other. The weapon which we wield so readily, but understand so little about. The basis of law, government and order in human society, and concurrently the basis for war, capitalist exploitation, and self-actualization.

The second path is, of course, insanity.

A different universe in the mirror

I was completely stumped when I came across the following question:

Why does a mirror reflect left and right, but not up and down?

I thought about this question for about 3 hours. I searched the internet for explanations, and I should warn you that practically all of them are incorrect, or miss the point entirely. It is almost an embarrassing question to think about. If I had been asked this question in person, I would probably have shot off a self-assured "explanation" out of vanity, before returning to the privacy of my room to reconsider it.

Many things dawned upon me as I contemplated what it all meant. My conclusion is that there is a different universe on the other side of the mirror. It is a universe of a distinctly different quality from ours. By this, I mean that we cannot mimic that universe without using a mirror (or something similar) in the first place. There are, in quite a literal sense, two universes, and the mirror is the conceptual impassable border between the two. Maybe I am completely off, so please let me know if you have a satisfactory answer to this question.

Of course, your reflection on the other side is contemplating the exact same thing as you are.

Dear SUSY... where are you?

What do bankers and theoretical physicists have in common?

Quite a lot possibly.

Theoretical physics has a certain public image, and quite a distorted one. There are fads, figureheads and fashionable fields. Undoubtedly the most in-vogue field is string theory, or its newer incarnation superstring theory, or the latest release, M-theory. Most people have heard something about string theory, either from watching the Big Bang Theory, or reading some popular science book by Stephen Hawking, or from general philosophical discussions about life, the universe, and everything.

It is really remarkable how the media and social environment dictates research even in a field like theoretical physics -- which purports to provide us with universal truth. String theory is, as far as I know, most dominant in the United States. It has the same sort of philosophy as American free-market capitalism. It has the vagueness of the American Dream. It has rock-star spokespeople that the young geeky types idolize. It also has the arrogance of American imperialism. It is also hated by more conservative physicists elsewhere.

I am not even exaggerating the extent of the divide that splits the world of theoretical physics. It goes much deeper, with shady businesses with government funding agencies and media moguls. What does this mean for this poor student who is navigating the murky waters in search for the right answers? Who should one believe, the confident string theorist whose mathematical prowess and alluring charm puts church pastors to shame, or the reticent, white-bearded professor who still speaks fondly of Einstein, Pauli, and Heisenberg?

I can't form an informed opinion with my modest understanding of string theory, and truth-be-told, there is a great fear of being left out on the deepest, most beautiful theory ever crafted. In all honesty, I have not been able to appreciate any of the depth in string theory, although there is purportedly a Holy Grail locked up somewhere in the minds of a company of select geniuses which one might gain access to after decades of dedicated study.

There are unsettling, and often unspoken problems at the root of theoretical physics. There are outlandish claims and false promises. There are deep philosophical questions left behind by Einstein that most physicists avoid. There is also my little issue with real numbers and their correspondence to reality. There is the irresistible irony of the brainwashed theoretical high-energy physicist who borrows sophisticated mathematical tools to build his gauge-field-theories-which-predicts-the-"God"-particle-now-build-me-my-Large-Hadron-Collider, but dismisses the nit-picky student who asks him what is so canonical about "canonical quantization." "We're doing, physics," he says, "not mathematics".

You will probably have heard about the Large Hadron Collider --  that atom-smasher that houses the most expensive experiment ever built. If you are slightly more interested, you will also know that it is supposed to nail down the Higgs boson (the God-particle I alluded to earlier on). You are probably wondering why this is such a big deal. If I were a particle physicist, I will regurgitate the gospel that it will validate out best fundamental theories, and bring us closer to our dream of finding GUTs (grand-unified theories) or TOEs (theories of everything). I will even extol the machine's unprecedented ability to find squarks, Higgsinos, and gluinos, particles that SUSY predicts.

As you nod your head politely while listening to my unintelligible attempts at justifying my existence, your most pertinent question is probably "Who is Susie?"

SUSY stands for "supersymmetry", which some might have you believe is the most beautiful feature of the Universe that its Creator endowed it with. There is only one problem. The LHC has not found it. And now, literally, theoretical physics is baring its ugly fangs as researchers declare war on each other's intellect.

It pains a little me to say this, but science is really quite a dishonest business. There is too little self-criticism, and too much peacockry. There is certainly nothing absolute about it, nor is anything ever "scientifically proven". Meanwhile, allow me to express my doubts about black holes, superstrings, multiverses, evolution, the Human Genome Project, natural selection, quantum computers, the Big Bang, probability and the scientific method, before I return to the 26-dimensional space-time of bosonic strings. 

Good music is back!

Today is Valentine's Day, as some have reminded me, thank you. But really, the most unexpected thing is that 3 of my favourite bands are suddenly in the news!

Firstly, there's the Strokes, or the Coolest Band in the World.

The hottest man in the world.

The coolest hair in the world.

And finally, the first Strokes album since 2006 is coming....on 21 March. Too bad for you if you didn't get the free download of "Under the Cover of Darkness" on Saturday. I was still in the army in 2006, what the hell!

Secondly, it's the shocking news about the Best Band in the World. The Arcade Fire upset Lady Gaga, Lady Antebellum (originality?), Eminem, Katy Perry to take the Grammy for Best Album of the Year.

Album of the Year. It's not cool to scramble for one now.

Did I say "Grammy"? Is an indie band allowed to win that? I have a feeling that they didn't really want the prize. After all, a Grammy slashes your cool factor by half. But I still love the pre-Valentine's Day Arcade Fire, and it will be interesting to see whether their next album will feature a shiny sticker on the cover that announces them as the "Winner of Best Album 2011".

Win Butler. How do these guys pull off such haircuts?

The "cereal guy", perpetually high.

While the rest of the world debates what this Grammy upset means for the direction of music, the Greatest Band in the World drops a bombshell. 

Yes, there really is a new Radiohead album.

What can I say. After 4 years, Radiohead has announced it's 8th album... to be released as a newspaper?!. This is fresh out of the oven today, and you can download the album digitally... this Saturday! So much for building up suspense and anticipation to boost your album sales. Too good for that. 

Proof that skinny guys are cooler than beefcakes.

The most pained man in the world.

I've already caught the Strokes in Scheeßel, and the Arcade Fire in Birmingham. So Radiohead must be next, and it will cost a lot... because they have a Grammy too (three in fact, for Best Alternative Album, but that's forgivable).

On a more sombre note, a part of me died when Arcade Fire won the Grammy. I am a vain bastard after all.

What's pageantry worth?

What do Liberians and Singaporeans have in common?

Absolutely horrid.

I also have stories about three extraordinary people... who hate getting prizes.

Grigori Perelman
Russian mathematican. Solved the Poincare conjecture, which I don't understand, but is apparently the greatest feat in mathematics for a long, long time. He was a shoo-in for the Fields Medal, probably the highest award in mathematics. As is quite well-known, he declined the prize. In fact, he did not even bother to send his work for publication. He just uploaded his proofs, and left the rest of the world in shock and awe.
He's extremely reclusive, unemployed (left his academic post), lives with his elderly mum and sister, and has more or less ceased to communicate with the rest of the world. It's a bit difficult to isolate yourself these days, so here's a picture of him taking the subway.

Perhaps he was just lazy to travel to Madrid to collect the prize. 
Says the man, "It was completely irrelevant for me...Everybody understood that if the proof is correct then no other recognition is needed."
Maybe a more practical prize would move him. After all, plane tickets from Russia are not very cheap. The Poincare conjecture happens to also be one of the seven Millennium Prize problems, i.e., the most devious mathematical problems known to humanity, which are worth $1,000,000 each. Actually, these problems are so difficult, the Clay Institute might as well have given a 1 gazillion dollar reward instead. Unfortunately for them, this crazy Russian did in fact solve the problem and they had to cough out the million dollars. However, Perelman, being himself, declined the money. In fact, he did not even attend the symposium that was held to celebrate his proof!


Alexander Grothendieck
Another mathematician whose work I have no inking about. Apparently, mere mention of his name in mathematical circles would trigger an atmosphere of intimidation. According to a story that I read some time ago, he had little worldly possessions, and would stay at friends' homes as he travelled around to attend conferences. Furthermore, he was an anarchist, and had strong political views. Many people claim to have political convictions, but Grothendieck went so far as to travel to the forests around Hanoi to deliver lectures on category theory as a protest during the Vietnam War, while the city was being bombed! He also abandoned a high-profile post created for him because it was funded by the military-industrial complex. 


Here's him (in shorts).

As is becoming mandatory among higher-level geniuses, he declined the Crafoord prize (although he took his Fields Medal). In he declination letter, he wrote, 

"...My salary as professor, even my pension starting next October, is more than sufficient for my own material needs as well as those of my dependents; hence I have no need for money. As for the distinction given to some of my work on foundations, I am convinced that time is the only decisive test for the fertility of new ideas, or views. Fertility is measured by offspring, not by honors...

...But is it not clear that superabundance for some is only possible at the cost of the needs of others?"

...agreeing to participate in the game of prizes and rewards would also mean giving my approval to a spirit and trend in the scientific world that I vew as being fundamentally unhealthy, and moreover condemned to disappear soon, so suicidal are this spirit and trend, spiritually and even intellectually and materially." 

He is still alive, but is stateless, and nobody really knows where he lives.

Richard Feynman
Everyone knows him, and he was not quite as crazy, but he still did his physics at a strip bar...

Perhaps coincidentally, they are all Jews.