Semistability, modular lattices, and iterated logarithms

TL;DR

This paper characterizes the asymptotic behavior of gradient flows on semistable quiver representations, revealing a recursive structure involving iterated logarithms and a canonical algebraic filtration within modular lattices.

Contribution

It introduces a recursive construction of approximate solutions and a new algebraic filtration applicable to any finite length modular lattice, advancing the understanding of asymptotics in gradient flows.

Findings

Asymptotics involve iterated logarithms.

A canonical algebraic filtration is identified.

Recursive solutions describe gradient flow behavior.

Abstract

We provide a complete description of the asymptotics of the gradient flow on the space of metrics on any semistable quiver representation. This involves a recursive construction of approximate solutions and the appearance of iterated logarithms and a limiting filtration of the representation. The filtration turns out to have an algebraic definition which makes sense in any finite length modular lattice. This is part of a larger project by the authors to study iterated logarithms in the asymptotics of gradient flows, both in finite and infinite dimensional settings.