Fixed Point Theorems a gentleintroduction Annual Seminar Week IITB
Overview• Some History• What are fixed points(FP)?• The statement of Brouwer’s Fixed point theorem o The hairy ball theorem• Coffee cup , hurricanes , and maps• An interesting construction of FP’s in a very restricted case
What are fixed points?• Fixed points to a functions are the points where f(x)=x• Fixed point theorem’s basically say that under certain conditions , f will have a fixed point• And variations in these conditions give rise to various fixed points theorems.
The obvious fixed point theorem• Every function that maps to itself in one dimension has a fixed point (a.k.a. the Intermediate-value theorem) x2 x1 x1 x2
Generalization to n-dimensions Brouwer’s fixed point theorem• Every continuous function from a closed ball of a Euclidean Space to itself has a fixed point.• Ball => Compact , Convex (not the spherical notion)• Euclidean Spaces -> n-dimensional spaces obvious examples are 2d spaces , 3d spaces
Hairy Ball theorem• You cant comb a hairy ball flat without creating a cowlick!
Some implications• Fixed point in a coffee cup!
Some implications• There is always a hurricane somewhere on the earth!• This follows from the hairy ball theorem and the fact that wind is a continuous transform .• Brouwer’s FPT is used by John Nash (“A beautiful mind”) to prove the existence of Nash-Equilibrium
Some implications• In computer graphics we sometimes need a continuous function that generates an orthogonal vector to a given vector.• The hairy ball theorem implies that there is no such function!
A Stronger FP theorem Kakutani Fixed point theorem• Constraint in Brouwer’s FP theorem is modified such that now the function is mapping to a subset of itself(closed ball).
Geometrical Construction offixed point, in a map overlay• Two maps of different sizes of a country are arranged on a table such that one of them lies on top of the other and is completely inside it.• FP in this setting would be a point on the table where both the maps point to the same location.• We use the fact that line joining FP to the vertices makes the same angle in both the cases with a corresponding edge of the map• And then some pure geometry !