Two Cute Proofs of the Isoperimetric Inequality

The blog has been pretty quiet the last few weeks with the usual end-of-term business, research, and A-exams (mine is coming up quite soon). I was looking through some of my notes recently and came upon two very short Fourier analysis proofs of the isoperimetric inequality. Both proofs are among my all-time favorites; the result is of general interest (though it is subsumed in more general and useful facts), and the proofs are quick and elegant. The proofs are similar, but the second generates a Poincare inequality which is one of the fundamental tools of analysis — basically, the inequality says that for a function with a derivative, the norm of the function minus its average value (this is known as a BMO norm) is controlled by the norm of its derivative.

Television Will Rot Your Brain

    A fun little math problem today.  It came about when I misheard a problem asked by a friend, and solved a slightly different problem.  It goes as follows:

    A child watches television everyday.  He always watches at least one hour a day, and he only watches tv in whole integer amounts of hours.  Concerned for his well-being, his parents impose a restriction: he can never watch more than 11 hours of tv in any 7 day period.  Show that there is some consecutive string of days in which the child watches exactly 20 hours of tv.

The number 20 is a relic of the problem that I misheard (show that this happens in any 11 week period).  Of course, why stop there?  Show that for any positive integer , there is some string of consecutive days in which he watches exact hours of tv.

    This begs an interest question: for what other ‘restrictions’ is this property true?  That is, what other numbers and have the property that if the child can only watch hours of tv in a day span, then he achieves every positive integer as a consectutive total?

Bad Talk Bingo: Brainstorming

    Next week I am off to IAS for the conference on Algebro-Geometric Derived Categories and their Applications.  I am starting to get worried about it, since the list of talk titles is up and only a few of them appear to be welcoming to non-experts in their respective fields.  Also, when registering, there was no option under ‘Occupation’ for ‘Grad Student’, which struck me as a subtle hint that I wasn’t welcome.  Perhaps just being at IAS will compel speakers to turn the difficulty up to eleven.

    In any event, I am preparing for there to be a fair share of bad talks.  This, and other recent experiences with bad talks has given me the following idea of something to do: Bad Talk Bingo.  The idea is to create a 5×5 grid of events and signs that a talk is going badly, so that I can check off which ones happen and try to get 5 in a row.  My plan is that if I ever succeed in getting 5 in a row, I am allowed to pretend to get a phone call and rush hurriedly from the talk.

Something Certain About Uncertainty

I was motivated by a comment on Jim Pivarski’s recent post to speak about the Heisenberg Uncertainty Principle. Someone asked,

If uncertainty in quantum mechanics comes from (or is inseparable from) quantization, then where does it come from in its mathematical formulation i.e in terms of a space and its Fourier transform?

The Heisenberg Uncertainty Principle is a curious fact: it requires no physical intuition whatsoever and yet has profound physical ramifications. It is also interesting because it is among a small group of facts which are both physically and mathematically interesting. It is an important (to harmonic analysis) and commonly known fact that a function and its Fourier transform cannot both be compactly supported. There are stronger statements than that, though, of the following flavor: if a function is a narrow spike near zero, then its Fourier transform will be a shorter and fatter bump around zero and vice versa. The Heisenberg Uncertainty Principle is a quantitative statement about this kind of fact.

Hyperbolic Discounting

    I have always been fascinated with the study of how people make decisions.  Its a complicated process that involves as much ‘gut instinct’ as rational evaluation, and is rife with systematic errors in judgement.  For a long list of common mistakes, check out Wikipedia’s list of cognitive biases, particularly the decision-making section.  The majority of them are things that most people are probably aware of, like picking a option just because other people picked it.  However, a couple of them are a bit surprising, especially the one I want to talk about today: the phenomenon of hyperbolic discounting.

Quantum Bears

Hello all! It has been a very long time since I last wrote; I have been going to and from CERN as we’re preparing for the LHC. I’m in Geneva right now, and I just came back from watching En Pleine Nature (Into the Wild). Without giving away too much plot, this movie contains a bear that did not eat anyone. It made me think of Grizzly Man from a few years ago, in which another bear did. What impressed me most about Grizzly Man is that the probability of being eaten by a bear, should we find ourselves face-to-face, is not a simple 20%. It depends a great deal on who the bear is, what he thinks about humans, how hungry he is, what I smell like, the weather, his mood, etc. There’s a whole space of parameters, and some regions of this space are filled with nearly 100%, others with nearly 0%.

Let’s say we’re doing a scientific study of bears eating people. In our first experiment, we put 100 people in the woods and just count how many get eaten. Then we’d like to get more grant money, so we do a more in-depth analysis by controlling for several variables: some of our volunteers are smeared in honey barbeque sauce, others aren’t. Our sequence of studies slowly sharpen the focus on the bear-eating parameter space, identifying the high-probability regions and the low-probability regions. Where does this process end? If we could do infinitely many studies, would we find that each point is either 100% or 0% (deterministic bears)? Or not (uncertain bears)?

Framing a discussion of quantum mechanics in this way illustrates a feature that is often missed: the connection between quantization and uncertainty. Usually these two topics just fall out of the postulates with little indication that they are related. As it turns out, quantization makes fundamental uncertainty possible.

Odd Sums of Consecutive Odds

Oscar Wilde’s character Algernon said in The Importance of Being Earnest, “One must be serious about something, if one is to have any amusement in life.” Of course in Wilde’s typical ironic fashion, Algernon was only referring to his own dedication to frivolous diversions. In that spirit, allow me a few moments to tell a story about one of the odder sums of odd integers I discovered as a kid.

I remember that sometimes when I was bored — most especially during long, bi-weekly car trips with my parents — I would play various games with integers. I have no idea why, but at one point I memorized some huge list of powers of 2 (I can still remember the list from 1 to 65,536). I also computed the squares, cubes, and so forth of most of the smaller integers. As a result, I discovered on my own quite a number of interesting patterns in the integers. I don’t remember most of them, but there is one in particular that has stuck with me through the years.

My Favorite Prime Number with Four Divisors

    A prank I recommend to readers is to use the number 91 when a group situation calls for a random prime number.  If done subtly enough, a decent portion of mathematicians will believe you.  Granted, its not a particularly funny prank…

Puzzles, Groups, and Groupoids

Over at Good Math, Bad Math, MarkCC has a nice post introducing groupoids which uses the fifteen puzzle as an example. I like this example a lot, and I thought it would be interesting to expand on it a bit. So I’m going to tell you:

1. Why the Rubik’s Cube is a finite group,
2. Why the fifteen puzzle is a finite groupoid, and
3. How to solve the fifteen puzzle.

I’m not going to assume any knowledge of groups or groupoids, but if you don’t know much group theory, you’ll have to skip over certain parts of the second half.

An Almost-Proof of the Four Color Theorem

I recently gave an Olivetti (our graduate colloquium) on chord diagrams, effectively covering my first two posts on the subject. In preparation for the talk, I read a little bit more about some cool things one can do with them, and I finally got around to reading a paper of Bar-Natan on connections with the Four Color Theorem. I figured I should write a post on it before everyone completely forgot what I’ve said already; that said, this post should be readable even if you didn’t read my other posts on chord diagrams.