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DTSTAMP:20260717T203334Z
SUMMARY:M. Petrache: Singular bundles and regularity of Yang-Mills minimize
 rs in supercritical dimension.
DESCRIPTION:Abstract:  I will present approximation\, existence and regula
 rity results for Yang-Mills minimizers in supercritical dimensions\, based
  on a joint project with Tristan Rivière. The starting point are Uhlenbec
 k’s results which provided the analytic foundations for Donaldson’s st
 udy of Yang-Mills connections on bundles over 4-manifolds. The object of s
 tudy in that case was the class of Sobolev connections on smooth bundles. 
 In dimensions 5 and higher the space of Sobolev connections over smooth bu
 ndles does not allow to apply the direct methods of the Calculus of Variat
 ions to obtain Yang-Mills minimizers. The substitute is a space of weak co
 nnections over singular bundles\, in which a weak closure result allows co
 nstructing Yang-Mills connections by direct minimization. This space is a 
 real measure-theoretic counterpart to singular objects of more algebraic f
 lavour\, e. g. coherent reflexive sheaves. The main tool for the optimal p
 artial regularity result for Yang-Mills minimizers in 5 dimensions is an a
 pproximation of weak connections by connections with finitely many topolog
 ical defects. Such approximation allows to apply a Morrey space analogue o
 f Uhlenbeck’s result\, relaxing the approximability hypothesis from prev
 ious singularity removal results by Tao-Tian and Meyer-Rivière. We will c
 ontrast this new approximation result with approximation results for nonli
 near Sobolev maps and for integral currents.
URL:https://www.imperial.ac.uk/events/107340/m-petrache-singular-bundles-an
 d-regularity-of-yang-mills-minimizers-in-supercritical-dimension/
DTSTART;TZID=Europe/London:20131107T140000
DTEND;TZID=Europe/London:20131107T150000
LOCATION:United Kingdom
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