Stuff upcoming

October 26, 2009

So it looks like I’ll be giving the Georgia Tech analysis seminar on Wednesday, November 18. It will be a talk about the power estimate from above on the Buffon needle probability of a needle landing in a 3^{-n}-neighborhood of a 1-dimensional Sierpinski gasket. The paper hasn’t been updated on the arXiv yet, but I’m confident that such an update is imminent, and I’ll post about it here. It will be readable, and an explicit exponent will be given.

From there, there’s a workshop at Wayne State the following weekend with a few colleagues. Then in January, there’s the Joint Mathematics Meeting in San Francisco, January 13-16, which I will catch at least part of.

I suppose I should get my details straightened out so I can actually get reimbursed.

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Brouwer

brouwer

Story: Man, no one knows what this dude’s deal is, but he’s friggin’ pugnacious. Armed with blunt force and sharp intuition, Brouwer never retreats from any battle. Unlike other villains, the very thorough Brouwer is never satisfied to leave his foes in the grip of a death machine, content in the knowledge that they can’t not die. To him, there is always another possibility, and he insists on being there when your last life runs out, and on inspecting the body himself. Only an equally-determined adventurer should dare offer opposition.

It’s also not known how such an unusual choice of characters got chosen to play the role of his sworn enemy, the Fubini Brothers:

fubinitonelli

On the left, Fubini, and on the right, the often-neglected Tonelli. While in reality Italians and turtles for the most part coexist peacefully, for the purposes of this game, they are sworn enemies.

Powers: Intuition. Brouwer is very hard to fake out, so you have no choice but to use your slight speed advantage to its optimum, dodging at the last second more often than not.

Also, simplicial decomposition. Sometimes, Brouwer likes to break the stage into little pyramids. has complete command over pyramids, so you should try to work quickly. If he completes the simplicial decomposition, then you’re pretty much sunk at that point.

Signature move: Find The Fixed Point. A magic crystal continuously moves things back and forth on the giant disc on which you fight him. Every time it does this, he uses his intuition to find a point which will not be disturbed by this tumult, and stomps down hard on it, claiming it as his own so as not to get tossed about like everyone else in the ensuing chaos, also demonstrating that he’s better than you because he actually knows where the fixed point is. Nyah, nyah.

Weaknesses: Excluded middles.

Strategy: First, annoy him really bad by telling him your favorite two-line proof that an irrational number raised to an irrational power is sometimes a rational number. This will throw him into a convulsive fit of angry stomping, in which he shows off his Find The Fixed Point power again and again, demonstrating the superiority of the constructive approach… or so he thinks. In the process, he opens up a hole in the floor.

Ending: Standing on neither one side nor the other, and there being no other conceivable platform on which to stand, he freefalls. You can only assume that there’s nothing off-screen to catch him, though Brouwer himself doesn’t rule out the possibility, offended at the thought that everything has to stand on one platform or its opposite in order to claim a share of existence. You’re almost certain that since Brouwer isn’t on either of the two platforms, then he simply does not exist anymore. Fubini and Tonelli walk off into the sunset satisfied that their work is done. The curtain falls, and then the message “the end?” appears on screen.

Man, that’s a really corny suspense cliche right there.

Rumor: If you play the game while your microphone is plugged in, and the music of Cantor is playing, then Brouwer gets even angrier. Truly, there are no depths of marketing gimmickry to which producers will not stoop these days. However, at least this is more in line with the nature of turtles. They really hate death metal. If you don’t believe me, find a turtle and some Slayer and try it yourself sometime in real life.

Everybody’s doing it

October 13, 2009

Dr. Volberg has a blog now, too. If anyone knows Russian and wants to make sure the jokes aren’t about me, that would be nice. I’m not too concerned for the most part, though the comic he’s currently running across the top has me slightly worried.

Cantor

cantor

Story: Though perhaps not a villain in the strict sense, there is no denying the wickedness that is Cantor’s jams. You might recognize him as the final boss of Karaoke Apocalypse, the monk with a slammin’ death metal sound that astounded punk rocker Mittag-Leffler, who declared that “it’s about one hundred years too soon for such righteous awesomeness to descend on the Earth.” Lead singer David Hilbert replied that Cantor death metal was here to stay, and that “No one shall expel us from the Paradise that Cantor has created!” At which point Cantor was crowned The Supreme Cardinal, which could never be exceeded. But the next move shocked everyone.

“Friends, rockers, fans, musicians,” Cantor addressed the screaming, enraptured masses. “Let it be known that there will always be a greater Cardinal! We must prove it… again and again, without end! I hereby declare a perpetual Karaoke Apocalypse!”

Who will be the next Greater Cardinal? Will it be you?

Powers: See for yourself:

Signature song: Continuum Hypothesis, probably the most awesome death metal song ever, but friggin’ impossible to sing because of a line that goes like this:

unsingable

…and then back down again in reverse. It’s questionable whether it’s even possible for a human voice to rise that rapidly, but then again, no one knows whether Cantor is a mere mortal.

Weaknesses: None, really. The final stage of Karaoke Apocalypse is pretty much friggin’ impossible. However, there is a choice to make. You have to win at Continuum Hypothesis if you want the good ending, but a lesser ending is available if you play the easier Power Set instead. In this ending, you become the next Greater Cardinal, but the next year, a new challenger defeats you by playing the Power Set twice, playing better than ever before the second time, and reaching a new, higher level of Cardinality. Your fame is short lived.

Rumor: While it is unknown whether any gamer has ever gotten the good ending without using a cheat code, Kurt Gödel claims that he has done it, but Paul Cohen argued that since it’s pretty much friggin’ impossible, no amount of evidence could ever conclusively prove that it had been done. But at the same time, if anyone could do it, it’s probably Kurt Gödel, and it’s equally impossible to prove that he didn’t do it, either.

Pro tip: Practice makes perfect! You might have to do the Power Set several times before you become the Greater Cardinal, so don’t give up!

Hausdorff

hausdorff

Story: Felix Hausdorff has always lusted for power. No source of knowledge has ever been too arcane for his dark curiosities, and no warning could ever dissuade him from his search as an incorrigible youth. Perhaps the most fateful day of all was the day in which he stumbled upon the dark tome of Sierpinski, in which he read of a powerful relic, Sierpinski’s Triforce:

sierpinski triforce

This relic has a special power: if covered by a set of open discs D_i with radius r_i, and the r_i^(log3/log2) sum to less than 100, then it grants n wishes, where n = log(1/r_j) and r_j is the largest of the r_i used in the covering. It is not known what dark purposes Hausdorff has in mind, but only a few pieces remain outside of his grasp, locked in the royal vaults of Leipzig. Until now, it had been unknown what strange purpose these otherwise useless fragments served, but now Hausdorff has begun his assault on the kingdom of Chain Rule in search of the others. He must be defeated before it is too late.

Powers: Can cover sets very efficiently. Unlike the wizard Lebesgue, who you fight earlier in the game, Hausdorff can detect you, even when you have shape-shifted into your Cantor Dust form. You won’t be able to sneak up on him, so don’t even try!

Signature move: T2. Hausdorff chooses two targets and conjures two magical bubbles around them. Nothing inside of one of the bubbles can have any effect on the inside of the other bubble. The targets must be disjoint and compact at the time of casting, or else the spell fails. There are exotic realms in which Hausdorff’s T2 special is ineffective, but you’d have to be a seriously l33t gamer to try that strategy.

Weaknesses: Nazis. However, you should stick to cleaner tactics if you want to get one of the good endings. One approach is to fight him in a pathological realm where his covering powers don’t work, but a more common tactic for novices is to get him to argue about Nietzsche, causing him to transform into his lesser form, Paul Mongré, for 30 seconds or so.

Rumor: It has been rumored that this is Hausdorff’s true form:

hausdorff-true form

But that’s just antisemitic propaganda.

Pro tip: When you fight him, make sure Banach and Tarski are in your party. Don’t finish him off until he casts the spell 2-sphere decomp. This will unlock their paradox signature move, to be learned at higher levels.

The 3^{-n}-neighborhood of the 1-dimensional Sierpinski’s gasket decays in Buffon needle probability at least as fast C/n^p, for some p>0.

Alexander Volberg will probably be posting our result soon. It’s like the last one, only better.

1. Spam bots troll this site like mad.

2. The filters do an insanely good job of catching them. So… good job.

I found this while fishing files out of an old computer the other week, and I figured you should have it. It’s all rather silly, but here’s my writeup of the secretary problem from 2004 or so. There’s a funny identity for partial sums of the harmonic series at the end, though, and it’s pretty bizarre. But it’s true.

It could probably have done just as well without the self-deprecating weirdness about girls being pretty and wishing they’d like me, but that’s where my head was at the time, and it is what it is.

There’s a discussion on Wikipedia as well.