Fixed Point theorems

In mathematics, fixed point theorems state that a function that obeys certain conditions will have at least a fixed point (a point such that \( f(x)=x \)). There are many of such theorems, but a reasonable amount of work for a group is to focus in the two simplest cases.

One of the most amazing connections of these fixed points theorems is in fact with “discrete” problems. This project will allow you to use the fixed point theorems to prove results about games. This will explore the conection of these fixed points with economics and game theory.

Brouwer fixed-point theorem

The one-dimensional case of this theorem is the most simple example. It states that a continous function on a closed interval \( [a,b] \) to itself must have a fized point. This is easy to prove and to visualize.

You will also need to at least understand what this theorem say in higher dimensions. The illustrations of the wikipedia page should all be understood and explained.

One of the most amazing consequences of Brower’s fixed point theorem is the fact that a board game, known as HEX can never end in a draw (see this), a fact first proved by the great John Nash and only published in an internal report.This fact can also be proved by other arguments, but the connection with Brouwer’s fixed point theorem is actually true in both ways: one can use the fact that the HEX game is a determined game to prove Brouwer fixed-point theorem. This was first published by David Gale in 1979.

Borsuk–Ulam theorem

The one dimensional example of this theorem has in fact been proved in class. It is a simple consequence of the intermediate value theorem, and was used to prove that two antipodal points on earth have the same temperature. There are versions of this theorem that work in higher dimensions, this is called Borsuk–Ulam theorem.

You will need to understand what this theorem says, and some of its consequences. In fact an even more surprising fact than the one shown in class is true, thanks to this theorem: There exist two antipodal points in earth where not only the temperature is the same, but also the pressure is the same.

You must have an idea of what the theorem says (also for dimensions higher than 1), and show some examples. The wikipedia page of the article is a bit technical, but contains excellent references. Among them, the great video tutorial about Borsuk–Ulam and the necklace splitting theorem.

The proposed connection with discrete problems is just the above mentioned necklace splitting theorem. You should understand the solution of the problem (assuming that Borsuk–Ulam is true).