yang-mills flow is one of the classical geometric evolution equations, having much in common both with harmonic map flow and with ricci flow. i will describe its key properties, focusing on the refined energy identities that have led to new results about the flow in dimension four and on special-holonomy manifolds.

the integral geometric satake equivalence in mixed characteristic
abstract: the geometric satake equivalence establishes a link between
two monoidal categories: the category of perverse sheaves on the local
hecke stack and the category of finitely generated representations of
the langlands dual group. it has many important applications in the
study of the geometric langlands program and number theory. in this
talk, i will discuss the integral coefficient geometric satake
equivalence in the mixed characteristic setting. it generalizes the
previous results of lusztig, ginzburg, mirkovic-vilonen, and zhu. time
permitting, i will discuss an application of this result in constructing
a jacquet-langlands transfer.