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Integrability meets Control Theory: Harmonic Maps in General Relativity

Connections for Women: Mathematical General Relativity September 03, 2013 - September 04, 2013

September 04, 2013 (09:00 AM PDT - 10:00 AM PDT)
Speaker(s): Shabnam Beheshti (Rutgers University)
Location: MSRI: Simons Auditorium
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC


Abstract We provide a framework for analyzing axially symmetric harmonic maps on R3 with symmetric target spaces G=K. Drawing on results from analysis to Lie theory to geometry, we give a complete and rigorous proof that, all such maps are completely integrable. We further demonstrate that new solutions to the harmonic map equations can be generated from a given seed solution, using a dressing or vesture method. This uni es the integrability of theories including chiral eld models, nonlinear -models, Yang-Mills and Einstein electrovacuum equations in the general context of harmonic maps. Utility of the vesture method is made concrete by generating N-solitonic harmonic maps into a noncompact Grassmann manifold G = SU(p; q). We demonstrate a special case by deriving Kerr and Kerr-Newman solutions from the Minkowski initial seed for the Einstein vacuum and Einstein- Maxwell cases, respectively. In performing an asymptotic analysis, these solutions are shown to be in the hyperextreme sector of the corresponding parameters, suggesting constraints on the dressing mechanism. We indicate the possibility of using this analysis to control the resulting N-black hole confi gurations in this setting.
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