The celebrated results of K. K.Uhlenbeck furnished the bases for a
complete variational study of the Yang-Mills functional in dimension
4, leading to the definition of new differential invariants of
4-manifolds by S. K. Donaldson. In this setting the objects of study
were Sobolev connections over smooth bundles. In the first part of the
talk I will show why it is not sufficient for the variational study of
the Yang-Mills functional in dimensions higher than 4.
In the second part of the talk I will define a new space of weak
connections introduced in collaboration with Tristan Rivière. I will
describe our existence-and-regularity theory for Yang-Mills minimizers
in dimensions greater than 4. More precisely, similar to the work of
Federer-Fleming on the Plateau problem, in our space of weak
connections at the same time we have existence and regularity of
stationary points for the Yang-Mills energy in dimension 5 and higher,
which are bundles with Yang-Mills connections and topological
singularities over a subset of the base manifold of codimension 5.
This gives a first step for connecting the variational theory to the
algebraic geometry framework which motivated some conjectures of Gang
Tian. The optimality of the codimension 5 is shown by a n explicit
example of a minimizer, and the proof of its minimality uses a new
combinatorial technique together with a decomposition theorem for
vector fields by S. Smirnov.
I will save some time at the end to present the natural next steps
in the study of these singular Yang-Mills bundles, namely the
conjectures of G. Tian and the rigorous construction of the
Yang-Mills-Gibbs measure from statistical mechanics.