The space of weak connections in general dimensions

We make a parallel between the study of connections over principal
G-bundles with low regularity assumptions, and that of harmonic maps,
in both cases using natural geometric energies. In higher dimensions
(>2 for harmonic maps or >4 in nonabelian Yang-Mills theory) vortices
of nontrivial degree are not energetically prohibited, and indeed
energy-minimizers are known to form topological singularities.
However while for maps these singularities are forming "holes in the
graph" of the functions and there is no change in the coordinates in
which the graphs have to be studied, for connections on bundles the
topology of the bundles "follows" by Chern-Weil theory the
structure/degree of the singularities of the connections, therefore in
general (as the singularities can form a dense set) there may exist no
good classical local coordinates.

In joint works with Tristan Riviere we tackle this problem by using
singular trivializations which "translate"  changes in the bundle's topology
into "jumps" of the trivialization maps, so that at first sight "any bundle
becomes trivial". The new feature of this class of "weak connections on
singular bundles" is that existence of minimizers always follows naturally.

I will describe the approximation and slicing methods which then
furnish a link to the classical Sobolev bundle regularity theory by
Uhlenbeck later extended by Isobe, Meyer, Riviere, Tao, Tian. For
minimizers these tools still allow to recover the classical smooth
bundle structure outside a codimension-5 singular set.