Abstract

Fractional Calculus, the theory of differentiation and integrationof non-integer orders, is a flourishing field of researchwith several physical applications such as modellingof anomalous diffusion and visco-elasticity.Despite its long history, the theory still lacked a comprehensiveand unified description for fractional derivatives and integralsacting on distributions on the real line.This was resolved using a quite general definitionfor the convolution of distributions,which goes back to Laurent Schwartz,but received almost no attention in the literature,appart from several experts specialized in the theoryof locally convex spaces of distributions.As an application, linear media with fractional dynamics,that are usually described as fractional initial value problems,are reinterpreted as convolution equationsof distributions on the whole real axes.
This successful application motivated studying a general classof locally convex spaces of distributions, called amalgam spaces.A subclass of these are the convolution perfect spaces,the natural largest domains associated to a set of convolution operatorsdefined by distributional convolution kernels.One particularly useful result about these spaces is,that the continuity of bilinear convolution mappings between such spacescan be characterized via their simpler discrete global components.
The main part of this talk will be preceededby a brief presentation of a hobby project by the author,that he intends to transform into a serious research project.One objective is the development of an efficient algorithmfor the inversion of the geodesic logarithm on Riemannian surfaces.The algorithm shall permit a varying base pointand an extension of its domain, not only beyond the injectivity radius,but also beyond the degeneracy radius.An application is the interactive visualizationof the intrinsic geometry of surfaces.