Let's start the topic "math explains world" (M.E.W.) with a result from Algebraic Topology. In its mathematical formulation the theorem looks not very interesting:
If f: S^2 ---> IR^2 is continuous, then there exists a pair of antipodal points x and -x, s.t. f(x) = f(-x), where S^2 denotes the 2-sphere (something which we can imagine as the surface of a ball) and IR^2 is the real number plane.
This theorem is the so-called Borsuk-Ulam-theorem. StanisĹaw Marcin Ulam, who also was involved in the early development of nuclear weapons, assumed that this theorem might be correct. Karol Borsuk, a Polish mathematician, proved 1933 the theorem.
But why should we be interested in such a boring thing from the last century?
Well, I can reassure you, it's not that boring. ;-)
The application of this theorem implies a result which is really mindblowing.
Let us consider temperature and pressure on earth. We can imagine the S^2 to be the surface of the earth. Antipodal points on earth are just points which lie on opposite sides of the S^2, like the north and south pole. There is a nice antipodal map on http://www.antipodesmap.com/, which helps to find antipodal points on earth. For example an antipodal point to Berlin would be somewhere in the south-west of New Zealand.
The map f could be a function that assigns points of the earth to a pair of two values, temperature and presssure.
If we can assume that this map is continuous, then the theorem says that there is always a pair of antipodal points on earth, which has the same temperature and pressure. Think about this! o.O
But why should this statement true? I will explain it.
!!! WARNING: The following section is written for freaks and nerds who are really interested in the idea of the proof. If you are not of this type, then you should skip the following. ;-) !!!
I will give only the idea of the proof by several pictures. This is not really mathematically correct, but intuitive.
Proof: In Algebraic Topology the term of covering spaces is connected with this theme (it's also connected with the term of the fundamental group). What is a covering space? We only have to know what is the covering space of the S^1 (unit circle or 1-sphere) to proof the theorem. The so-called universal covering of the S^1 is a map from p: IR ---> S^1 which has the following properties and is shown in the modified picture from Hatcher's Algebraic Topology book (2001, p. 29).
A small (open) neighbourhood U (in red) of an arbitrary point of the S^1 has a bunch of corresponding (open) neighbourhoods in IR (also in red), s.t. we can map them to the first neighbourhood by p and p induces a homeomorphism from one of the neighbourhoods to U, this means that the neighbourhoods are really âsimilarâ to the neighbourhood U. In the following proof we can assume that p(t) = e^(2 pi i t) (cf. https://en.wikipedia.org/wiki/Euler%27s_formula). The proof follows Hatcherâs proof (2001, p. 32,33).
Now we are well prepared for the proof. We prove the theorem by assuming the contrary and leading it to a contradiction (https://en.wikipedia.org/wiki/Proof_by_contradiction).
Let's suppose that such a map exists, but has no point x, s.t. f(x) = f(-x). Then we can define the function g: S^2 ---> S^1 with g(x) = (f(x) - f(-x))/|f(x) - f(-x)|. We can calculate just by inserting -x in the formular, that g(-x) = g(x). We call the equator loop (cf. picture down) y: [0,1] ---> S^2, where we choose x_0 as the basepoint of the loop, i.e. 0 and 1 get mapped to x_0 in S^2.
This is indeed a loop, because it starts and ends at the same basepoint x_0. The composition g*y given by g*y(x) = g(y(x)) is a map from [0,1] to S^1 which starts and ends at the same point, hence a loop in S^1.
Since we know that a neighbourhood of a point in S^1 has a bunch of neighbourhoods in IR by the property of the covering space, we can construct a unique path in IR by lifting pieces of the loop g*y in S^1 by this lifting-property of the covering space to IR. This path is unique, if we fix basepoints, i.e. we map 0 in IR to 1 in S^1 by p. (Rmk.: Consider that 1 is in S^1, because we are working in the complex plane. If you are not firm working with complex numbers, just read (1,0) instead of 1 and consider the real number plane IR^2.) We can construct this lifting of the loop in finitely many steps (by compactedness). We call this lifting G*Y which is just a path from [0,1] to IR. So we have G*Y: [0,1] ---> IR. Now we can compare loops in S^1 with paths in IR. An example of the path G*Y in IR (in blue) is given by the following picture:
The corresponding loop at the base S^1 is also coloured blue (modified picture from Hatcher, 2001, p. 29).
Let's consider the difference d: [0,1/2] ---> IR given by
d(t) = G*Y(t) - G*Y(t + 1/2) = G(Y(t)) - G(Y(t+ 1/2)).
This difference is just the difference of antipodal points in S^1 in terms of the difference of the corresponding paths in IR. We know, if we go once around the circle S^1, i.e. we have a loop starting and ending at 1 in S^1 (see Rmk. before), then we go from 0 to 1 in IR, like we see it in the picture. If we go 1/2 of the way of the loop g*y in S^1, then we have two antipodal points z and -z in S^1. Now we consider p: IR ---> S^1 given by p(t) = e^{2 pi i t}. Any two antipodal points z and -z in S^1 give integers x in p^{-1}(z) and x' in p^{-1}(-z) which differ just by a number of the form "n + 1/2" where n is an integer. Looking at the preimage p^{-1}(z) or p^{-1}(-z) means just looking at the bunch of points in IR above the points z and -z. But being something of the form "n + 1/2 = (2n + 1)/2 = odd integer/2" means just that the difference
d(t) = G*Y(t) - G*Y(t + 1/2) = q/2, where q is an odd integer.
  G*Y(1) - 0 = G*Y(1) - G*Y(0) = G*Y(1) - G*Y(1/2) + G*Y(1/2) - G*Y(0)
  = (G*Y(1) - G*Y(1/2)) + (G*Y(1/2) - G*Y(0)) = q/2 + q/2 = q.
So G*Y(1) = q not equal to 0 and this means that G*Y starts in 0 and ends in q. But this is a contradiction, because y is homotopic to the constant loop c that stays for its whole interval [0,1] at one point (i.e. c(t) = x_0), i.e. we can deform y in S^2 continuously to a point, like in this picture:
But then g*y is homotopic to a constant loop, hence G*Y, too. But we saw that G*Y doesn't stay at one point, but goes from 0 to q. So it can't be the constant loop. CONTRADICTION!
Next time we will look at an interesting consequence of the theorem, which has something to do with a division of the world. But this will not be written before next month. See you then!
First picture: https://de.wikipedia.org/wiki/Karol_Borsuk#/media/File:Borsuk.jpeg, https://de.wikipedia.org/wiki/Stanis%C5%82aw_Marcin_Ulam#/media/File:Stanislaw_Ulam_ID_badge.png
Second picture: https://de.wikipedia.org/wiki/Erde#/media/File:The_Earth_seen_from_Apollo_17.jpg
Third and fifth picture: Modified pictures from
Hatcher, Allen (2001): Algebraic Topology, URL: http://www.math.cornell.edu/~hatcher/