Analysis on fractals

TLDR: From Manifolds to Fractals Analysis on manifolds has been one of the central areas of mathematical research in the twentieth century as discussed by the authors, and it has attracted mathematicians with diverse expertise and points of view, including topology, differential equations, differential geometry, functional and harmonic analysis and probability theory.

Abstract: From Manifolds to Fractals Analysis on manifolds has been one of the central areas of mathematical research in the twentieth century. Rooted in the foundational work of the nineteenth century, with its rigorous theory of multidimensional calculus and the visionary ideas of Riemann, it has flowered into a richly layered mathematical tapestry. It has attracted mathematicians with diverse expertise and points of view, including topology, differential equations, differential geometry, functional and harmonic analysis, and probability theory. This heady mix of ideas has produced a vast body of work and a seemingly endless supply of challenging problems that should keep mathematicians busy well into the next century. At the same time it has become apparent that many phenomena in the real world are best modeled by geometric structures that are much more irregular. The theory of fractals, as B. Mandelbrot [Ma] has so forcefully argued, seeks to provide the mathematical framework for such development. A theory of analysis on fractals is now emerging and is perhaps poised for the kind of explosive and multilayered expansion that has characterized analysis on manifolds. This article will explain some of what has been accomplished and where it might lead. The central character in the theory of analysis on manifolds is the Laplacian. Thus the starting point for analysis on fractals will be the construction of an analogous operator on a class of fractals. This will not be a genuine differential operator, of course, but it will have quite a few of the features we have come to expect from anything labeled “Laplacian”. It will be a local operator, and in fact ∆f (x) will be a limit in a suitable renormalized sense of the difference between an average value of f in a neighborhood of x and f (x). We will be imitating the weak formulation of the Laplacian, so that ∆u = f will be interpreted to mean