Towards the GSS conjecture

As is well-known to experts, the classification of minimal
surfaces of Class VII with positive second betti number is
not yet complete. The Global Spherical Shell Conjecture is
widely believed to be true, and an affirmation of this
conjecture would complete the classification. Andrei
Teleman has introduced a strategy based on gauge theory for
attacking the problem, and in remarkable work, he has used
this to completely resolve the case of second betti 1 and
partially resolve that of second betti number 2. To extend the
method even further will require the overcoming of some very
significant obstacles. In this talk, I shall describe some
ideas that might form the skeleton of a different approach
to tackling the conjecture.