Blow up for Stationary Yang Mills

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• Aaron Naber (Northwestern University)
• Thursday 06 October 2016, 15:30-16:30
• CMS, MR15.

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Given a principal bundle P-> M over a Riemannian manifold, let us consider a stationary Yang-Mills connection A with curvature F and bounded energy. If we consider a sequence of such connections A_i, then it is understood that up to subsequence we can converge A_i-> A to a singular limit connection. As is standard in nonlinear pde, the convergence may not be smooth, and we can understand the blow up by converging the energy measures |F_i|^{2 dv_g → |F|}2 dv_g +\nu, where \nu=e(x)d\lambda^{n-4} is the n-4 rectifiable defect measure. It is this defect measure which explains the behavior of the blow up. It has been an open problem to compute e(x) explicitly as the sum of the bubble energies which arise from blow ups at x, a formula known as the energy identity. This talk will primarily be spent explaining in detail the concepts above, with the last part focused on the recent proof of the energy identity, joint with Daniele Valtorta.

This talk is part of the Special DPMMS Colloquium series.

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