Morse Spectra, Homology Measures and Parametric Packing Problems

TL;DR

This paper explores mathematical analogs of effective degrees of freedom, inspired by Guth's work on Morse spectra and homology in the context of volume energy functions on cycle spaces, aiming to understand asymptotic behaviors.

Contribution

It introduces new mathematical frameworks relating Morse spectra, homology measures, and packing problems, inspired by Guth's asymptotic results.

Findings

Proposes several mathematical counterparts to effective degrees of freedom.

Formulates specific questions on Morse (co)homology spectra.

Draws inspiration from Guth's results on asymptotics.

Abstract

We suggest several mathematical counterparts to the idea of "effective degrees of freedom" and formulate specific questions, much of which are inspired by Larry Guth's results and ideas on the Hermann Weyl kind of asymptotics of the Morse (co)homology spectra of the volume energy function on the spaces of cycles in balls.