John Palmieri

Fixed points, antipodal points, ham sandwiches, and hairy balls: some classical theorems in topology

Topologists study shapes and properties of them which don't change if you bend, shrink, stretch, or twist them. Gluing or punching holes is not allowed, though. For example, a cube and a sphere are topologically equivalent, but these are not topologically equivalent to a doughnut or a coffee cup. We'll discuss some classical theorems in topology, starting with the Brouwer fixed point theorem and the hairy ball theorem, and moving on to the Borsuk-Ulam theorem and the ham sandwich theorem. Aside from having good names, these theorems will let us explore some basic topological concepts and techniques. We will also use some results from combinatorics to prove them.