Monday, May 27, 2013

Inter-universal Geometer Mochizuki

Mathematics certainly does not lack for eccentric geniuses. Mochizuki homepage. See these slides from his "pedagogical" lecture Invitation to Inter-universal Teichmuller Theory  :-)
The Paradox of the Proof: ... The problem, as many mathematicians were discovering when they flocked to Mochizuki’s website, was that the proof was impossible to read. The first paper, entitled “Inter-universal Teichmuller Theory I: Construction of Hodge Theaters,” starts out by stating that the goal is “to establish an arithmetic version of Teichmuller theory for number fields equipped with an elliptic curve…by applying the theory of semi-graphs of anabelioids, Frobenioids, the etale theta function, and log-shells.”

This is not just gibberish to the average layman. It was gibberish to the math community as well.

“Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space,” wrote Ellenberg on his blog.

“It’s very, very weird,” says Columbia University professor Johan de Jong, who works in a related field of mathematics.

Mochizuki had created so many new mathematical tools and brought together so many disparate strands of mathematics that his paper was populated with vocabulary that nobody could understand. It was totally novel, and totally mystifying.

As Tufts professor Moon Duchin put it: “He’s really created his own world.”

It was going to take a while before anyone would be able to understand Mochizuki’s work, let alone judge whether or not his proof was right. In the ensuing months, the papers weighed like a rock in the math community. A handful of people approached it and began examining it. Others tried, then gave up. Some ignored it entirely, preferring to observe from a distance. As for the man himself, the man who had claimed to solve one of mathematics’ biggest problems, there was not a sound.

... When Mochizuki posted his papers, the math community had much reason to be enthusiastic. They were excited not just because someone had claimed to prove an important conjecture, but because of who that someone was.

Mochizuki was known to be brilliant. Born in Tokyo, he moved to New York with his parents, Kiichi and Anne Mochizuki, when he was 5 years old. He left home for high school, attending Philips Exeter Academy, a selective prep school in New Hampshire. There, he whipped through his academics with lightning speed, graduating after two years, at age 16, with advanced placements in mathematics, physics, American and European history, and Latin.

Then Mochizuki enrolled at Princeton University where, again, he finished ahead of his peers, earning his bachelor’s degree in mathematics in three years and moving quickly onto his Ph.D, which he received at age 23. After lecturing at Harvard University for two years, he returned to Japan, joining the Research Institute for Mathematical Sciences at Kyoto University. In 2002, he became a full professor at the unusually young age of 33. His early papers were widely acknowledged to be very good work.

Academic prowess is not the only characteristic that set Mochizuki apart from his peers. His friend, Oxford professor Minhyong Kim, says that Mochizuki’s most outstanding characteristic is his intense focus on work.

“Even among many mathematicians I’ve known, he seems to have an extremely high tolerance for just sitting and doing mathematics for long, long hours,” says Kim.

Mochizuki and Kim met in the early 1990s, when Mochizuki was still an undergraduate student at Princeton. Kim, on exchange from Yale University, recalls Mochizuki making his way through the works of French mathematician Alexander Grothedieck, whose books on algebraic and arithmetic geometry are a must-read for any mathematician in the field.

“Most of us gradually come to understand [Grothendieck’s works] over many years, after dipping into it here and there,” said Kim. “It adds up to thousands and thousands of pages.”

But not Mochizuki.

“Mochizuki…just read them from beginning to end sitting at his desk,” recalls Kim. “He started this process when he was still an undergraduate, and within a few years, he was just completely done.” ...
In other news, "unheralded" mathematician Yitang Zhang proves the (weak) twin primes conjecture. Remarks by his PhD supervisor at Purdue; Zhang was reputed to be one of the top mathematics students in his class at Beijing University, so, despite his difficult career path, he was not exactly an unknown quantity.

I have only limited interest in number theory (for some reason it just does not get me excited), but unlimited admiration for people like Mochizuki and Zhang (and Perelman and so on). I think someone once described Wiles' proof of Fermat's last theorem as a "triumph of the human spirit" -- I could not agree more!


Mulcaster said...

"...working for several years as an accountant and even in a Subway sandwich shop..."

The free market at work. Gotta love it.

Mulcaster said...

"I think someone once described Wiles' proof of Fermat's last theorem as a
"triumph of the human spirit" -- I could not agree more"

That's very disheartening. If one believes g is more than jive great mathematicians are wasting their talents.

Here's a "triumph of the human spirit" and the most important result in the natural sciences in 50 years or more:

Mulcaster said...

Really this result makes all of theoretical physics, at best, an intellectual circle jerk.

Pseudonym said...

I have a certain amount of admiration for people like Mochizuki, but I have a lot of exasperation, too.

Mathematics is a conversation. It is "done" by individuals, but it is also "done" by the community. You have not achieved a result unless the community believes that you have. It is, therefore, your job as a mathematician to explain your work to the community, especially if it is difficult to understand.

Mochizuki has shirked his responsibility as a mathematician. This is not admirable.

Chris Adami said...

I have a problem with ascribing the "genius" moniker to someone incapable of explaining their ideas. If you truly understand something, you can explain it. I have read many of Einstein's papers, and they are clear, intuitive, and beautiful. (Of course it helps that I read them long after they were written). Same is true for Schroedinger, less so Heisenberg. Bohr, on the other hand... Don't get me started. Biut I also have read several of Ed Witten's papers, and he is just as clear as Einstein. For me, a red flag goes up when I read stuff that either was written for the purpose of obfuscation, or when the purpose is to make the writer look god-like. There is no need for this in science or math. If you figured something out, you ought to be able to explain it.

steve hsu said...

You are right about this, although perhaps his upcoming colloquium in June at the University of Tokyo might signal some increased willingness to explain the work?

steve hsu said...

Yes, Mochizuki is an oddball among oddballs. It's not even clear his claimed proof is correct. I should have been more specific in stating that my admiration is for people who are willing to work for years and years on very ambitious projects. Perhaps Zhang, Perelman and Wiles are better examples ...

David Coughlin said...

I used to think this dogmatically, If you truly understand something, you can explain it. I agree less and less [though I won't say that I disagree, it just becomes less and less relevant]. Intellectual explorations are, by nature, personal exploits. When you take on and make progress in hard, obscure subjects the community has some obligation to follow, rather than you having some obligation to lead. Finding that division of labor is rarely [never?] negotiated explicitly in an intellectual community. I would guess that it is mostly tribal knowledge, and tribes are not usually friendly to outsiders.

Mulcaster said...

It's amazing what a tiny penis will drive a man to do.

5371 said...

As your own examples show, the capacities are closer to being orthogonal.

Mulcaster said...

"...Mochizuki’s most outstanding characteristic is his intense focus on work."

Work? "Laziness at doing stupid things is a great virtue."

Pseudonym said...

I think it's fair in Mochizuki's case. The mathematical community knew he was a genius before they knew he was working on this.

Indeed, he'd probably be ignored as a crackpot if he didn't already have established credentials. There's some precent; consider Srinivasa Ramanujan and Stephen Wolfram. Only time will tell which of the two Mochizuki is closer to.

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