Alexander Friedrich
The solution spaces of problems in geometric and global analysis often consist of unions of (Banach-)manifolds with varying dimensions which makes analysis difficult and cumbersome. Classical examples include the bubbling of harmonic maps and holomorphic curves.
M-polyfolds and scale calculus were developed to provide a framework for these problems. They generalize classical differential geometry and can provide an alternative smooth structure to stratified topological spaces with dimension jumps.
In this talk we present a brief introduction to M-polyfolds and discuss an example with dimension jump in detail.
Additionally, we discuss how flows on M-Polyfolds with dimension jumps can be used as autoencoder models in machine learning. To this end, we recall the key concepts of autoencoders and neural ODEs. We close by presenting prove of concept geometric reconstruction experiments.
