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.

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