We introduce the notions of path and homotopy, and give some different ways of thinking about them.

We introduce homotopy equivalence of spaces, and show that \(\mathbb R^n\) is contractible.

An important example of homotopy equivalence is when a space deformation retracts onto a subspace.

We explain the idea behind spectral sequences, and give an example.

A brief tutorial in Grapher, a useful OS X program for visualizing and exploring problems in calculus and geometry.