As I understand it, it’s a branch of mathematics dealing with knots and wadded-up pieces of paper. I’m confident that it IS cool and useful, I just don’t know (at present) how or why.
Anybody wanna fight my ignorance? (Though I warn you, it’s a substantial opponent…)
We use certain aspects of topology in the electronic mapmaking business. Sometimes it’s important to be able to answer such questions as:
What’s connected to what
What is adjacent to what
How is this map feature related to this other feature
And at the bottom level those are all topological questions.
a coffee cup can become a donut
I don’t know as much about topology as some, but what I do know is that you can use topology to prove that a donut is the same shape as a cup. Just push and pull that donut until it’s got a depression in one side and is flat on the bottom, and there you go.
Is that not cool?
I wouldn’t call it dealing with knots and wadded up paper – I’d call it the maths dealing with surfaces, which is how you can prove mathematically that said donut is the same shape as said cup (even though you’re on your own if you try to get the donut to hold your coffee).
A donut and a coffee cup has the same topology as a human too!*
*Super-over-simplified. But cool.
Am I the only one who has actually used topology in my (non-academic) work?
Some of the applications I’ve seen mentioned:
The computer you’re typing. Squeezing more and more transistors into smaller and smaller spaces of nanometers uses topology research to optimize how circuit trace lines are layered in 3D PCB boards in consideration of heat, interference, etc.
Sending up a spaceship (or spacelab) can unfold a giant solar panel. The collapsed fan would fit very compactly in the rocket. (This is where the knowledge of wadded up pieces of paper is used.)
Determining the minimum skin graft (triangular size and shape) required to cover a burn wound of a particular shape.
It seems like application of topology is infinite…
The one-line, non-jocular answer: topology is the study of continuous functions (ways of associating input values with output values in such a way that inputs that are really close go to outputs that are really close).
What’s any of this got to do with shapes and such things? Well, to formalize this business about continuous functions, you need to say “A continuous function from what to what?”. The whats are not just collections of possible values, but also some extra structure upon them which you can think of as what you need to interpret the “Continuous functions preserve real closeness” business. And with this extra structure, you can think of the values as being points in some space, forming some shape, the rough connectedness properties and so on all being described, but concepts like “Is this a straight line? Is this twice as long as that?” not being defined.
But, again, in one line, modern topology is really the study of continuity.
I know the joke about how a topologist is a guy who doesn’t know the difference between a cup of coffee and a donut, and I understand the concept (both are volume-occupying forms with a single hole through them).
What I don’t get is how you can turn that observation into math, and I especially don’t get how you can use it to solve real-world problems whose answer isn’t obvious by looking. Can anybody give a specific example of such a problem?
But of course a donut is not the same shape as a cup… on some (dare I say, the usual) definition of “shape”.
Topology is about a very special notion of “shape”. One that ignores almost all features. What features doesn’t it ignore? The features preserved by continuous functions: the idealized description of how some parts of the figure are really close to some other parts of the figure. Everything else is thrown away.
That’s why I think it’s best to emphasize from the start that topology (and by this, I mean only the particular field modern mathematicians refer to as “topology”, and not necessarily any other ordinary language applications of the term) isn’t necessarily about “shape” as one is inclined to think, but really fundamentally about the idea of continuous variation.
If you’re a little bit kinky, telling someone you’re interested in topology actually qualifies as a double-entendre. Not a great one, but still. 
The version I prefer is that a topologist is someone who can’t tell his ass from a hole in the ground, but who can tell his ass from two holes in the ground.
But for practical applications, here are a few results that come out of topology theorems:
At any given time, there exists a pair of antipodal points on the Earth which have both exactly the same temperature and the exact same barometric pressure. (an application of the Intermediate Value Theorem)
Given a four-legged table at some location on an uneven surface, it’s always possible to rotate the table so that it doesn’t wobble. (another application of the Intermediate Value Theorem)
At any given moment, there is always at least one point on the surface of the Earth where the wind speed is exactly zero. (the Hairy Ball Theorem)
This notion of the number of holes something has really isn’t as essential to topology as all the jokes would have you believe. I think Indistinguishable’s explanation is best: topology is the study of functions where changing the input by a little means that the output only changes by a little (and vice versa).
As for examples, topology is such a central area of modern mathematics that it’s honestly a little tough to think of problems where topology doesn’t have an application. Topology has applications in robotics, programming language design, economic theory, image processing, network design, and the modeling of any complex system you care to think about, as well as many other things.
Well, certainly, (assuming continuous variation of such things), the IVT would let us get a pair of antipodal points with exactly the same temperature, and similarly a pair of antipodal points with exactly the same pressure. In fact, we could get such “opposite points with same values” pairs along any loop we like, for any particular single-dimensional quantity we like. But to get both temperature and pressure to be the same simultaneously? This would seem, to me, to require tools more sophisticated than the IVT (the Borsok-Ulam theorem, for example).
[The example is of some nostalgic significance to me because I have a very clear memory from high school of being told just this statement, trying to prove it from the IVT (after a sudden breakthrough in biology class on the simple way to do so for the 1-dimensional version), struggling, and finally thinking I could munge it together with the aid of certain “intuitive, but somewhat dodgy” leaps of logic, forcing me to ponder “Hm… Are these leaps legitimate or not? They seem rightish… but then, there’s no point in having something merely proofish, is there?”. Indeed, the leaps were ultimately untrue. I still don’t know if there’s a nice way to get it from just the IVT, though I’d be interested to see one.]
Typos:
A) In editing, I forgot what I originally introduced scare quotes for. I meant to punctuate as follows: “intuitive”, but somewhat dodgy
B) It’s spelt Borsuk.
Take such a loop of antipodal points for, say, temperature. Now apply the IVT for pressure, along that loop. Done.
How did you get a loop of antipodal points for temperature?
You take the temperature along every point on that loop. Somewhere in there, the temperature goes up, and then comes back down. At some point, the temperature at the antipodes is equal per the Intermediate Value Theorem. Actually, you can use any quality you like–pressure, altitude, population within a radius, partial pressure of radium gas, whatever–as long as the quality can be plotted as a continuous curve that is differentiable at all points.
Stranger
I don’t think you understood my question. I understand how to use the IVT to determine that every continuous function from a circle to a line sends a pair of antipodal points to the same value. Chronos however is going further then this: he is stepping up from the 1-dimensional case to the 2-dimensional case. He proposes to use the IVT to determine that, for every pair of continuous functions from the surface of a sphere to a line, there is some pair of antipodal points which have this property simultaneously with respect to both functions. As the first stage of his proof, he apparently obtains a loop of points all of which have the same temperature as their antipode [with this loop also, presumably, going through at least one pair of antipodal points]; as the second stage, he just applies the 1-dimensional version of the theorem to find, among this loop, a point whose pressure is equal to that of its antipode as well.
My question is, how does he pull off the first stage? How does he guarantee himself that a full continuous loop of points will exist, all of which have the same temperature as their antipodes [and with the loop also running through at least one pair of antipodal points]? Once he has done this, I agree that it is trivial to satisfy the pressure constraint as well; it’s how to do this first stage which I cannot see.
