A1-contractibility and topological contractibility

These notes are an extended transcript of two lectures given at the University of Ottawa during the workshop “Group actions, generalized cohomology theories, and affine algebraic geometry." The first lecture was a general introduction to A1-homotopy theory focusing on motivating the choice of definitions in the construction, together with a definition of A1-contractible spaces. The second lecture attempted to understand A1-connectedness in greater detail, and to use this to understand better the relationship betweenA1-contractibility and topological contractibility. I have taken the liberty of including some material in these notes that I hoped to (but was unable) to discuss in the lectures. In particular, I have included some problems which I felt might be interesting to different groups of participants. Disclaimer: these notes were not carefully proofread; please e-mail me with any comments, corrections or questions.