Dear Colleagues,

In 1975, J. Harrison conjectured the existence of a duality between differentiability classes of functions and fractal dimensions of domains, incorporating classical operators of geometry, including boundary and Hodge star. This would come in the form of a directed system of domains, paired with a downward directed system of differential forms. The smoother the functions, the rougher the domains could be. Hints of such a dichotomy have appeared in her counterexamples to conjectures of Denjoy and Seifert and an underlying theory began to take shape. 

The directed system is not apparent when looking at currents, especially when considering operators, but becomes clear when turning the variance around. Whitney believed that methods based on domains were primary, so that, in particular, the boundary operator could be defined directly, not as the dual to the exterior derivative. However, he did not have the entire directed system of domains. His sharp and flat chains of geometric integration theory came close, but neither captured the required properties. It was not until differential chains were introduced that a directed system of domains was found solving the conjecture. This theory gives analysis on the fractals in a natural setting and includes generalizations of the integral theorems of calculus to nonsmooth domains with nonsmooth boundaries. Classical operators of geometry, such as boundary, Hodge star and pushforward, have direct definitions on polyhedral approximations and are continuous.

Harrison’s 1993 “Stokes' Theorem for nonsmooth chains” and a closely related paper 1992 “A Gauss-Green theorem for fractal boundaries” marked the beginning of this work and have influenced other papers. There has been a recent resurgence of interest in analysis on nonsmooth or fractal domains.

A goal of this special edition is to present research papers relating the differentiability class of functions and the dimension of domains.

Prof. Dr. Jenny Harrison
Guest Editor

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• unsmoothable diffeomorphisms
• topologically conjugate to a diffeomorphism
• dimension and differentiability
• differential chains theory
• chain complex of Banach spaces of polyhedral chains
• nonlinear differential equations
• functional analysis
• differentiability class of functions
• the dimension of domains