These draft notes are from a graduate course given by the author in Berkeley
during the spring semester of 2005. They cover the basic ideas of a new,
geometric approach to geometric measure theory. They begin with a new theory of
exterior calculus at a single point. This infinitesimal theory extends, by
linearity, to a discrete exterior theory, based at finitely many points. A
general theory of calculus culminates by taking limits in Banach spaces, and is
valid for domains called ``chainlets'' which are defined to be elements of the
Banach spaces. Chainlets include manifolds, rough domains (e.g., fractals),
soap films, foliations, and Euclidean space. Most of the work is at the level
of the infinitesimal calculus, at a single point. The number of limits needed
to get to the full theory is minimal. Tangent spaces are not used in these
notes, although they can be defined within the theory. This new approach is
made possible by giving the Grassmann algebra more geometric structure. As a
result, much of geometric measure theory is simplified. Geometry is restored
and significant results from the classical theory are expanded. Applications
include existence of solutions to a problem of Plateau, an optimal Gauss-Green
theorem and new models for Maxwell's equations.