I'm presenting Brouwer's fixed point theorem to an audience that knows some point-set topology. Does anyone have any zippy / enlightening / cool applications or consequences of it? So far, I have:
Thanks! :)
The theorem is equivalent to the determinacy of the Hex game. That's a very famous `application'.
The details can be found in [David Gale (1979). "The Game of Hex and Brouwer Fixed-Point Theorem". The American Mathematical Monthly 86: 818–827], a beautiful paper, which JSTOR serves at http://www.jstor.org/stable/2320146.
The existence of mixed Nash equilibriums in multiplayer games can be proved using the Brouwer fixed point theorem. If you imagine that each players is trying to improve his results based on the current actions of his opponents, then there is some combination of strategies so that each player can't improve.
Just for fun you could give them the 2-dimensional version of the standing-in-a-train theorem. That theorem says that if you want to go to sleep while standing up in a train that goes along a perfectly straight track, there is some starting angle that will cause you not to fall over. (Proof: if the angle is almost all the way forwards, then you will fall forwards; if it is almost all the way backwards, then you will fall backwards; by the intermediate value theorem there must be an angle that leaves you still standing at the end.) Ian Stewart argues vigorously that the implicit continuity assumption (that your final position depends continuously on your initial position) is wrong, but you can just forget about that.
Now let's suppose that you are standing on a surface that can move horizontally in any direction. (Perhaps you are in a boat, say.) This time if you fall, your head will be somewhere in a circle centre your feet and radius your height. Assuming that the position you end up in depends continuously on your starting position, this defines for you a continuous map of the disc that preserves the boundary. By Brouwer it is not a retraction, so there must be a starting position that stops you falling over.
Even if the continuity assumption is not in the end justified, I think this is a good and amusing illustration of the theorem.
The Brouwer Theorem can be used to prove that a mapping of ${\bf R}^n$ to itself that has bounded displacement, in the sense that any point is moved at most a fixed amount from its original location, is onto. This seems be a folklore result. I wonder if anyone has a reference for it.
In economics and game theory, it's used to prove the existence of equilibrium (eg Nash equilibrium, general equilibrium)
By the fixed point theorem, one can prove that a polynomial with complex coefficients has at least a root in complex plane. It is not really cool but fits your audience.
The following is not exactly an application, but rather a funny picture to illustrate the theorem, precisely in the form of the non-retraction theorem.
Suppose a shark jumps into a shoal of fish (a kind of big ball). The small fishes start escaping in all directions towards the border of the shoal, where the fishes stand still. Yet they escape with a certain disposition to follow a continuous flow, as they usually do, since everybody tends to follow its neighborhoods. But since there is no continuous retraction to the boundary, somebody doesn't know where to go, and stay there for a moment, much to the shark's satisfaction. There is also a 2D version, with a wolf entering into a herd of sheeps. This is just a funny picture, though I like to think that there is some truth in it.
In many nonlinear equations, the existence of a solution (but not its uniqueness) follows from a topological argument in the vein of BFP Theorem. The BFP is at work especially when the equation is posed in some finite dimensional vector space, and you can establish an a priori estimate of the size of the solution. This means that you know a ball $B_R$ containing all the solutions.
The most important example of this situation is the stationary Navier-Stokes equation, with Dirichlet condition $u=0$ on the boundary of the domain. Of course, the ambient space is infinite dimensional, so you first establish the existence of an approximate solution in a subspace of dimension $n$ (Galerkin procedure); this is where you use the BFP Theorem, or its equivalent form that a continuous vector field over $B_R$ which is outgoing on $\partial B_R$ must vanish somewhere. Then passing to the limit as $n\rightarrow\infty$ is pedestrian.
The BFP Theorem is a consequence of the fact that the Euler-Poincaré characteristic of the ball is non-zero. There are counterparts when you work on a compact manifold (with boundary) whose EPC is non-zero. This happened to me in a very interesting way. I considered the free fall of a rigid body in water filling the entire space. The mathematical problem is a coupling between Navier-Stokes and the Euler equation for the top. I looked at a permanent regime, in which the solid body has a time-independent velocity field, and that of the fluid is time-independent as well, once you consider it in the moving frame attached to the solid. The difficulty is that you don't know a priori the direction of the vertical axis (the direction of gravity) in this frame. After a Galerkin procedure, the problem reduces to the search of a zero of a tangent vector field over $B_R\times S^2$. This vector field is outgoing on the boundary $\partial B_R\times S^2$. Because $$EP(B_R\times S^2)=EP(B_R)\cdot EP(S^2)=1\cdot2\ne0,$$ such a zero exists. Therefore the permanent regime does exist. Remark that because $EP=2$, we even expect an even number of solutions when counting multiplicities, at least at each level of the Galerkin approximation.
The Schauder fixed point theorem can be proved using the Brouwer fixed point theorem. It says that if $K$ is a convex subset of a Banach space (or more generally: topological vector space) $V$ and $T$ is a continuous map of $K$ into itself such that $T(K)$ is contained in a compact subset of $K$, then $T$ has a fixed point.
The Schauder fixed point theorem in its turn is an important tool for existence proofs in differential equations. One easy application is the Peano existence theorem, but there are also more sophisticated examples.
I really like the short proof of Jordan curve theorem using Brouwer’s fixed-point theorem given in these lecture notes.
One standard consequence of Brouwer's theorem is Borsuk's antipodal mapping theorem, which in turn is used to prove that if $E$, $F$ are subspaces of a normed space and the dimension of $E$ is strictly less than the dimension of $F$, then there is a unit vector in $F$ whose distance to $E$ is one. This result is used often in Banach space theory.
The proof of the fundamental theorem of algebra using the Brouwer fixed point theorem is given here:
B.H. Arnold, "A topological proof of the fundamental theorem of algebra" Amer. Math. Monthly , 56 (1949) pp. 465–466
It can be used to show that a matrix with positive entries has an eigenvector with positive entries: that shows the existence of an invariant measure in the theory of finite space Markov chains.
The Perron-Frobenius theorem (in various degrees of generality) is an easy consequence of Brouwer's fixed-point theorem.
I've been thinking about a very similar issue: I was considering giving a talk about the Brouwer fixed point theorem to some math majors (but not necessarily ones very familiar with point-set topology).
There is an elementary proof which uses Sperner's lemma; see Michael Henle's book A Combinatorial Introduction to Topology for details. You can outline a proof of Sperner's lemma pretty quickly (induction on dimension, and dimension 1 is easy), and from that, you can wave your hands (or be more precise, depending on how much your audience knows about compactness) to get Brouwer's theorem. Since Sperner's lemma holds in higher dimensions, you can prove the fixed point theorem in higher dimensions, also.
Simple and example, for which You may perform demonstration here, is to use some piece of rubber balloon and a lamp.If You stretch it and show that it leaves a shadow on the table. Then leave rubber to fall on the table, it should drop within area where the shadow was dropped previously. There should be point which after falling ( and release of stretching) should remain in precisely the same position.