Abstract
We consider applications of the theory of balanced weight filtrations and
iterated logarithms, initiated in arXiv:1706.01073, to PDEs. The main result is
a complete description of the asymptotics of the Yang--Mills flow on the space
of metrics on a holomorphic bundle over a Riemann surface. A key ingredient in
the argument is a monotonicity property of the flow which holds in arbitrary
dimension. The A-side analog is a modified curve shortening flow for which we
provide a heuristic calculation in support of a detailed conjectural picture.