Math Translator Wanted/Topic Map Needed: Mochizuki and the ABC Conjecture

What if you Discovered the Answer to a Famous Math Problem, but No One was able to Understand It? by Kevin Knudson.

From the post:

The conjecture is fairly easy to state. Suppose we have three positive integers a,b,c satisfying a+b=c and having no prime factors in common. Let d denote the product of the distinct prime factors of the product abc. Then the conjecture asserts roughly there are only finitely many such triples with c > d. Or, put another way, if a and b are built up from small prime factors then c is usually divisible only by large primes.

Here’s a simple example. Take a=16, b=21, and c=37. In this case, d = 2x3x7x37 = 1554, which is greater than c. The ABC conjecture says that this happens almost all the time. There is plenty of numerical evidence to support the conjecture, and most experts in the field believe it to be true. But it hasn’t been mathematically proven — yet.

Enter Mochizuki. His papers develop a subject he calls Inter-Universal Teichmüller Theory, and in this setting he proves a vast collection of results that culminate in a putative proof of the ABC conjecture. Full of definitions and new terminology invented by Mochizuki (there’s something called a Frobenioid, for example), almost everyone who has attempted to read and understand it has given up in despair. Add to that Mochizuki’s odd refusal to speak to the press or to travel to discuss his work and you would think the mathematical community would have given up on the papers by now, dismissing them as unlikely to be correct. And yet, his previous work is so careful and clever that the experts aren’t quite ready to give up.

It’s not clear what the future holds for Mochizuki’s proof. A small handful of mathematicians claim to have read, understood and verified the argument; a much larger group remains completely baffled. The December workshop reinforced the community’s desperate need for a translator, someone who can explain Mochizuki’s strange new universe of ideas and provide concrete examples to illustrate the concepts. Until that happens, the status of the ABC conjecture will remain unclear.

It’s hard to imagine a more classic topic map problem.

At some point, Shinichi Mochizuki shared a common vocabulary with his colleagues in number theory and arithmetic geometry but no longer.

As Kevin points out:

The December workshop reinforced the community’s desperate need for a translator, someone who can explain Mochizuki’s strange new universe of ideas and provide concrete examples to illustrate the concepts.

Taking Mochizuki’s present vocabulary and working backwards to where he shared a common vocabulary with colleagues is simple enough to say.

The crux of the problem being that discussions are going to be fragmented, distributed in a variety of formal and informal venues.

Combining those discussions to construct a path back to where most number theorists reside today would require something with as few starting assumptions as is possible.

Where you could describe as much or as little about new subjects and their relations to other subjects as is necessary for an expert audience to continue to fill in any gaps.

I’m not qualified to venture an opinion on the conjecture or Mochizuki’s proof but the problem of mapping from new terminology that has its own context back to “standard” terminology is a problem uniquely suited to topic maps.

Comments are closed.