  These lectures fall into two distinct, although tenouously related, parts.
The first part is about fuzzy and noncommutative spaces, and particle mechanics
on such spaces, in other words, noncommutative mechanics. The second part is a
discussion/review of twistors and how they can be used in the calculation of
Yang-Mills amplitudes. The point of connection between these two topics,
discussed in the last section, is in the realization of holomorphic maps as the
lowest Landau level wave functions, or as wave functions of the Hilbert space
used for the fuzzy version of the two-sphere. This article is based on lectures
presented at the conference on Higher Dimensional Quantum Hall Effect and
Noncommutative Geometry, Trieste, March 2005, Winter School on Modern Trends in
Supersymmetric Mechanics, Frascati, March 2005 and the
Montreal-Rochester-Syracuse-Toronto Conference 2005, Utica, May 2005.
