Avscon the Schur topology

TL;DR

This paper introduces an operator-theoretic topology for homogenisation processes, enabling convergence analysis and homogenisation results for time-dependent PDEs without relying on specific coefficients.

Contribution

It develops a novel operator topology that captures homogenisation convergence purely through operator-theoretic means, simplifying analysis of PDE homogenisation.

Findings

Operator topology fully captures homogenisation convergence.

Homogenisation results for time-dependent PDEs obtained almost automatically.

Provides a new perspective on homogenisation using operator theory.

Abstract

The aim of the course is to lead to an understanding of homogenisation processes in an operator-theoretic sense. In fact, using solely operator-theoretic means not referring to the particular form of the coefficients, we will identify an operator topology on the level of coefficients that will fully capture the convergence involved in the context of homogenisation. One upshot of this perspective will be that we will obtain homogenisation results for time-dependent partial differential equations (almost) for free.