Dynamics, Spectral Geometry and Topology

TL;DR

This paper explores the relationships between dynamics, topology, and spectral geometry on closed manifolds, focusing on counting rest points, instantons, and closed trajectories for generic vector fields.

Contribution

It provides an informal report on joint work analyzing how dynamical features relate to topological and spectral properties of manifolds, with emphasis on generic vector fields.

Findings

Counting of rest points, instantons, and closed trajectories for generic vector fields.

Connections established between dynamical features and topological invariants.

Framework applicable to a broad class of vector fields.

Abstract

The paper is an informal report on joint work with Stefan Haller on Dynamics in relation with Topology and Spectral Geometry. By dynamics one means a smooth vector field on a closed smooth manifold; the elements of dynamics of concern are the rest points, instantons and closed trajectories. One discusses their counting in the case of a generic vector field which has some additional properties satisfied by a still very large class of vector fields.