Semi-stability and iterated logarithms

A very important tool in the study of algebraic vector bundles is the notion of (semi-) stability. It turns out that any semistable vector bundle admits a canonical filtration whose subquotients are direct sums of stable bundles. This weight-type filtration depends only on the lattice of vector bundles of the same slope contained in a given bundle and is defined in the general context of modular lattices. It has an analytic interpretation in terms of certain gradient flows and this is how we originally discovered it. Conjecturally, the filtration describes the asymptotics of Donaldson's heat flow on the space of metrics on a semistable bundle. All this is joint work with Katzarkov–Kontsevich–Pandit (1706.01073).