Some recent work in Frechet geometry

TL;DR

This paper reviews recent advances in Frechet geometry, including the structure of second tangent bundles in infinite-dimensional manifolds and the hypercyclicity of operators on Frechet spaces.

Contribution

It extends finite-dimensional results to infinite-dimensional Frechet manifolds and summarizes new findings on operator hypercyclicity.

Findings

Extended second tangent bundle structure to infinite-dimensional manifolds

Characterized second tangent bundles and differential equations in Frechet spaces

Summarized recent results on hypercyclicity of operators

Abstract

Some recent work in Frechet geometry is briefly reviewed. In particular an earlier result on the structure of second tangent bundles in the finite dimensional case was extended to infinite dimensional Banach manifolds and Frechet manifolds that could be represented as projective limits of Banach manifolds. This led to further results concerning the characterization of second tangent bundles and differential equations in the more general Frechet structure needed for applications. A summary is given of recent results on hypercyclicity of operators on Frechet spaces.