A better yodeling contest

…although this concept doesn't seem to catch when I try explaining it to people. I've given the following explanation many times:

• game theory = economics
• combinatorial game theory = games

Granted, this is a somewhat simplistic view, but it does provide a good general idea. In any case, it doesn't seem to help.

Today, I was accosted in the hallway near my office to help someone work through a mixed-strategy (simultaneous move) game. Fortunately, it was simple enough I eventually knew how to do it, but I'm beginning to be worried people will start taking me as a fraud due to my inexpertise in the area, despite it not actually being my field. On a separate occasion this afternoon, someone assured me that a Nobel Prize in Economics had been awarded for "combinatorial game theory"; and again today, another individual emailed me a payoff matrix expecting my familiarity with it.

I suppose the problem of explaining your technical field is fairly common. Perhaps I needn't worry too much, since I probably know more about standard game theory than anyone prone to making this kind of mistake; however, seeing the same confusion three times in one day suggests I need to adopt a new strategy to deal with the situation.

I wouldn't mind learning enough game theory in the economic sense to keep up appearances, but I probably don't have time for that in the near future. So, I think I have no choice but to lie about the name of my field. Any mention of "games" will immediately be linked to popular game theory, no matter how much explaining I do. CGT could be seen as a subfield of "recreational mathematics", but then I might need to explain that it is, in fact, professional research. I might be able to get away with "combinatorics", or make up a name like "combinatorial, two-actor decision-tree analysis". What do you think?

(note: post written when tired, explaining the lacks of sense made)

Hi, please pardon the lack of posting on this blog while I attempt to understand Stokes (generalized) Theorem:
\[
\int_M \text{d}\omega = \int_{\partial M}\omega
\]

There is a bit going on behind the scenes. Like the Fundamental Theorem of Calculus, this is saying that the values of a derivative in a region is related to the values at the region's boundary; however, Stokes' Theorem does this for arbitrary manifolds in however many dimensions you prefer.

I understand it well enough to derive Green's Theorem and the Fundamental Theorem of Calculus. Now, I need to figure out how to derive the Divergence Theorem and Stokes (ungeneralized) Theorem before October 6, when my book is due.

It's called "All the Mathematics You Missed: But Need to Know for Graduate School". Fortunately, the other chapters cover topics that I used to know, so this might be all I missed.

Hopefully, I will soon find time to write the post entitled "You Think You Can Do Everything", which is naturally about cycling, trespassing, and fatherhood. See you then.

I ran across this post about automatic differentiation, which is a technique that can be used to numerically find derivatives. It uses something called dual numbers which are defined much like complex numbers, as pairs of reals. Instead of having an element $i$ such that $i^2=-1$, one has $d\neq 0$ so that $d^2=0$. Then one finds that:
\[f(x+d)=f(x)+df'(x)\]
for reasons that are fairly clear. "Aha!" I thought, one can generalize this idea to define a whole family of rings, where one takes the real numbers and adjoins an element $j$ where $j^2=\alpha+\beta j$. It turns out there are only 3 of these rings up to isomorphism: the complex, dual, and perplex* ( $j^2=1$) numbers, determined by whether $4\alpha+\beta^2$ is (respectively) less than, equal to, or greater than zero. Only the complex numbers are closed under arbitrary exponents - indeed, the duals and perplexes are not even fields, meaning division is not necessarily defined.

This seems to call out for iterating functions on the dual and perplex planes ala the Mandelbrot set, for the very sober mathematical purpose of discovering if one can create similarly pretty pictures.

Oh, while I'm linking to A Neighborhood of Infinity, I must point out this post in which you can read for yourself why $1+2+3+4+5+\cdots = -1/12$, and the product of all primes is $4\pi^2$. A similar regularization of a divergent sum provides another justification for the use of two's complement in computer systems, for then the quantity
\[\cdots11111_2=\sum_{n=0}^\infty 2^n = \frac{1}{1-2}=-1\]
as expected!(?)

(* The perplex numbers are usually known as the split-complex numbers, which sounds much duller in my opinion.)

What is the shape of our universe? We know that when we look around us, we see 4 dimensions: three are of course length, width, and height; the fourth is time. We also believe that you will see 4 dimensions basically wherever and whenever you go in the universe.

Knowing this, however, isn't enough to know the overall shape of the universe. The surface of the Earth has 2 dimensions, for instance latitude and longitude, but this alone is not enough to know its shape. A two-dimensional surface could be round like a ball, flat like a paper, looped like the surface of a donut, or even worse: it could be twisted in hyperspace so that if you went around the world, you would come back to where you started, but everything would be backwards.

How much harder is it to figure out the overall shape of the 4D universe? Much harder! If one has a complete atlas of the Earth, it is no hard work to determine that it must be roughly spherical. However, even if one had an atlas of the universe, it has been proved that no computer program could be guaranteed to tell you whether it is "simple" like a 4D ball, or has "handles" like a donut.

For those with some knowledge of abstract algebra or computer science, there is a recent paper that gives a beautiful overview of the proof that this problem is undecideable. The proof relies on two cool facts I'd never seen before, which give an idea of how the proof goes. First is that every finite group is the fundamental group of some 4-manifold - one can construct a manifold directly from the group representation. Secondly, any Turing machine can be encoded into a group, and an input mapped to an element which is trivial if and only if the Turing machine halts!

\[e=\sum_{n=0}^\infty \frac{1}{n!}\]

For those unsure of what exactly mathematicians do, Ars Mathematica has answered several of your questions. A highlight:

Question: Mathematics was my least favorite subject in high school/college.

Answer: I'm not an expert in communicating in English (not enough Greek letters), but I believe that this is not actually a question. But trust me, mathematics was our least favorite subject in high school and college too. But it wasn't until graduate school that we learned exactly how much you can hate a subject, and yet still keep taking classes in it.