One Less Problem to Solve this Week

Nets Katz is smarter than you, and he knows it.  (photo: IU)
Let's start off our week feeling stupid together, shall we?

IU mathematician credited with solving one of combinatorial geometry's most challenging problems

BLOOMINGTON, Ind. -- A mathematician in the Indiana University College of Arts and Sciences is being credited with resolving a 65-year-old problem in combinatorial geometry that sought to determine the minimum number of distinct distances between any finite set of points in a plane.
I know that math is hard, but the idea that any person or group of people would hammer away on a single problem for 65 years is staggering.  How do you show your work for that?  Or does the professor just accept your final answer since you're the first one ever to get it right?
The work by IU Department of Mathematics Professor Nets Hawk Katz, with Larry Guth of the Institute for Advanced Study in Princeton, N.J., achieved what many thought was unachievable: Solving Paul Erdös' 1946 Distinct Distances Problem.
First things first:  if "Nets Hawk Katz" isn't the most bad-ass name in mathematical history, I'd like to see what is.  And if there's ever a real-world use for "combinatorial geometry" maybe it's to create a machine that spits out names like Nets's.
"If someone hands you some distinct set of points, you can figure out what is the set of differences. The problem is to determine what the minimum possible set of distances is," Katz said. "What we did is to show that no matter how you place the N points, the number of distances is at least a constant times N/log N."
No, Nets, if someone hands you some distinct set of points, you can figure all that out.  If someone hands it to me, the last thing I'm going to do is divide it by the log of itself.  And you should know that, because it took you all this time to think of it.  What chance do I have?  And what is this "N" anyway?
Here, N represents the number of the finite set of points. In a historical achievement, Katz and Guth were able to reach the exponent of 1, heretofore thought to be impossible.
OK, understood.  I suspect this press release is soft-selling the work involved here, though.  I mean, who among us hasn't reached the exponent of 1 in their daily life?  You'd think they could at least have tried this just to see what happens.  Maybe even try it, I don't know, first since it's a relatively low number.  But, as they say, the exponent you're looking for is always in the last place you look.  Probably because once you find it you stop looking.
Building upon decades of work by others, Katz and Guth examined the problem within a group of rigid motions of the plane and used Euclidean geometry to view the distance problem as a three-dimensional linear one. Introducing two new ideas to the existing proof, the pair exponentially improved upon the most recently described lower bound of .8641, a mark Katz had previously helped obtain.
So, it is a lot more complicated than just trying the power of 1 to see how it feels.
Guth and Katz were able to combine the algebraic method with a result from topology called the polynomial ham sandwich theorem, which was used to create a cell decomposition that yielded the desired results when most points were in the interiors of the cells, while the alternative case could be handled by the algebraic method.
OK, now they're just making stuff up to sound smart.  The word "ham sandwich" appearing in the name of a mathematical operation tipped me off.
Fields Medal winner and UCLA math professor Terence Tao called the work "impressive" and the foundation for further advances.
We all saw Good Will Hunting, so we know the Fields Medal is a really big deal.  You don't earn one by just doodling away on an MIT chalkboard when you should be mopping the halls.  It takes hard word, arrogance, and the persistent use of a scarf to earn one of those, so Dr. Tao's endorsement of this work really means a lot.
"Now that we know that two of the most powerful tools in combinatorial incidence geometry -- the cell decomposition and the polynomial method -- can be combined to give nearly sharp results that were out of reach of each of the methods separately, it seems worthwhile to revisit all of the other standard problems in the subject and see if we can advance the partial results for these problems a bit more," he wrote of the newly discovered dichotomy during a review of the proof.
There's something fishy here.  I always thought of math as producing precise results.  You know, like 2 + 2 = 4.  You can joke that 2 + 2 = 5 for very large values of 2, but I know better than that.  When a Fields Medal winner claims that this breakthrough gives us "nearly sharp results" it sounds to me like "pretty close to accurate" or "only slightly approximate" and, again, I thought math was better than this.  Oh, well...
Janos Pach, editor in chief of the journal Discrete and Computational Geometry and one of the most frequent collaborators with Erdös, in a personal blog described the paper as "a great achievement." "Let us celebrate this fantastic development," he wrote.
Yes!  Let us celebrate.  But how best to do that?  Maybe a cash prize would be appropriate.  In this day and age of overpaid celebrities and athletes, it would be nice if someone doing the real work got a little remuneration for their efforts.  I mean, a horse can make millions just for running fast with a small man on its back, so would it be asking too much to hand a mathematician a big bonus check for solving a 65-year-old puzzle that is going to revolutionize an entire field?  As always, Paul Erdös was several steps ahead of us...
Erdös created a $500 prize for anyone coming up with the solution to the distance problem, which has long considered one of the most challenging problems in geometric combinatorics. Until now that prize had gone unclaimed. But at a recent meeting of the Canadian Math Society, Ron Graham, chief scientist at the California Institute for Telecommunications and Information Technology and the person in charge of the prize since Erdös' death, agreed after an audience discussion that the discovery warranted a $250 prize.
Are you *%$# kidding me!?  All this work, and he earns half of the $500 prize that was set aside years ago? And what has Ron Graham been doing with the money all this time?  He couldn't have put it in a CD or something to earn a little interest?  Some mathematician.  Even Einstein, who allegedly didn't have the wherewithal to remember his own phone number, would've done something with the money.  Fortunately, Katz speaks up to give Graham a little slack:
"The sense in which our result is non-optimal is only in the logarithms," Katz noted. "The lower bound is of the order of N/(log N) instead of the optimal N/(log N)^{{1/2}}, but the exponent -- which is the power of N in the numerator -- is the sharp exponent 1."
Oh, well, when you put it that way, I see your point.  Or, set of points, so to speak.