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Theme for TIFR Centre For Applicable Mathematics, Bangalore

Abstract: Defect of compactness for an embbedding of two Bamach spaces $$E\hookrightarrow F$$ is a difference between a sequence and its weak limit in E, up to a remainder vanishing in $$F$$ For many embeddings, defect of compactness can be expressed as profile decomposition - a sum of asymptotically decoupled terms. Profile decompositions refine the better known Concentration Compactness Principle of P.-L.Lions. We discuss profile decompositions generated by non-compact invariances, such as translations and scaling invariance, as well as the recent results on profile decompositions without invariance.