We are gald to announce the upcoming talk given by Tillmann Kleiner in the Oberseminar "Geometrie und Visualisierung" (Geometry and Visualization).

Title
Fractional Calculus on Distribution Spaces

Date, Time and Place
16.05.2024, 2:15 pm, Arnimallee 6 - Room 108/109

Abstract

Fractional Calculus, the theory of differentiation and integration of non-integer orders, is a flourishing field of research with several physical applications such as modelling of anomalous diffusion and visco-elasticity. Despite its long history, the theory still lacked a comprehensive and unified description for fractional derivatives and integrals acting on distributions on the real line. This was resolved using a quite general definition for 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 theory of 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 equations of distributions on the whole real axes.

This successful application motivated studying a general class of 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 operators defined by distributional convolution kernels. One particularly useful result about these spaces is, that the continuity of bilinear convolution mappings between such spaces can be characterized via their simpler discrete global components.

The main part of this talk will be preceeded by 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 algorithm for the inversion of the geodesic logarithm on Riemannian surfaces. The algorithm shall permit a varying base point and an extension of its domain, not only beyond the injectivity radius, but also beyond the degeneracy radius. An application is the interactive visualization
of the intrinsic geometry of surfaces.