Finding Fixed Points With Invariants -
Given a set X and a function f from X to itself, how can we determine if there is a fixed point, some point where f(x)=x? If X is a vector space, then one can get a complete answer by knowing all of the eigenvalues, but one can also get quick partial information by looking at the trace. If X is a topological space, what tools are at our disposal? In this talk, we will introduce topological spaces and an invariant called the fundamental group. We will use this tool to show that certain continuous functions ALWAYS have a fixed point, and draw a conclusion that every map (like a real map from an atlas) has a "You Are Here" point.