From linear spaces to metric spaces

The overarching goal of this proposal is to develop certain tools and applications of metric space geometry. One important notion is that of a metric functional, an extension of linear functionals.

The principal tools are results on the behavior of iteration of distance non-increasing maps and random compositions of such.

We are motivated by the analogy with linear functional analysis, with its weak topology, weak compactness, spectral and ergodic theorems.

Remarkably one can say that there are metric analogs of these features, and strikingly there are also powerful tools in this fiercely general setting.

The more immediate objectives are:- To contribute to the theoretical understanding of deep learning, espceilly the role of depth in neural networks.

Joint work with Benny Avelin, Uppsala,- To establish operator-valued multiplicative ergodic theorems, partly motivated by important evolution equations in infinite dimensions (such as differential equations modelling climate development),- To prove an analog of the Oseledets theorem in the setting of surface homeomorphisms extending Thurston´s spectral theorem.

Joint work with Camille Horbez, Paris-Saclay,- To establish a general metric fixed point theorem and study connections to the invariant subspace problem.The first step is to hire a doctoral student in order to advance these objectives. Avelin and I will submit a first paper during 2021. Horbez and I will finish the main theorem in 2021-2022 and then turn to applications.