Lectures on chainlet geometry - new topological methods in geometric measure theory

TL;DR

This paper introduces a novel geometric approach to geometric measure theory using chainlet geometry, extending exterior calculus to complex domains like fractals and soap films, simplifying classical results and enabling new applications.

Contribution

It develops a new chainlet-based framework that generalizes classical geometric measure theory with minimal limits, incorporating a richer Grassmann algebra structure for broader applications.

Findings

Unified theory for manifolds, fractals, and soap films

Simplified proofs of classical geometric measure results

New models for Maxwell's equations

Abstract

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.