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SUMMARY:The Yang-Mills Lagrangian in supercritical dimensions - Mircea Pet
 rache (ETH)
DTSTART:20130211T150000Z
DTEND:20130211T160000Z
UID:TALK43245@talks.cam.ac.uk
CONTACT:Prof. Neshan Wickramasekera
DESCRIPTION:The small energy regularity for Yang-Mills solutions in dimens
 ion 4 was the\nkey to the concentration-compactness result for Yang-Mills 
 fields by Karen\nUhlenbeck. This result\, proven about 3 decades ago\, was
  relevant for the\nconstruction of 4-manifold invariants by Simon Donaldso
 n. The study is\nbased on the theory of Sobolev connections over smooth bu
 ndles. I will\ndescribe in what sense dimension 4 is critical for Uhlenbec
 k's approach.\n\nI will then introduce a notion of weak curvatures on poss
 ibly very singular\nbundles. Such class allows to apply a variational appr
 oach to the\nYang-Mills Lagrangian in superctitical dimensions\, where we 
 don't have a\npriori control on the topology of the bundle. This measure-t
 heoretic notion\nof bundles allowing wild topological singularities bears 
 analogies to\nintegral currents. The integer multiplicity condition of Her
 bert Federer\nand Wendell Fleming is replaced in our setting by the integr
 ality of\nrelevant Chern classes.\n\nI will present the solution to the Ya
 ng-Mills-Plateau problem in the\nabelian case. In particular I will descri
 be a closure theorem for abelian\nweak curvatures on singular bundles and 
 the successive regularity result\nfor the Yang-Mills-Plateau minimizers. T
 hese results are part of an ongoing\nproject in collaboration with my advi
 sor Tristan Rivière.\n
LOCATION:CMS\, MR11
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