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Prescribing scalar curvature in high dimension

Andrea Malchiodi
October 2, 2018

We consider the classical problem of prescribing the scalar curvature of a manifold via conformal deformation of the metric, dating back to works by Kazdan and Warner. This problem is mainly understood in low dimensions, where blow-ups of solutions are proven to be "isolated simple". We find natural conditions to guarantee this also in arbitrary dimensions, when the prescribed curvatures are Morse functions. As a consequence, we improve some pinching conditions in the literature and derive existence
results of new type. This is joint work with M. Mayer.

Tensor rank

Avi Wigderson
Herbert H. Maass Professor, School of Mathematics
October 2, 2018
Tensors occur throughout mathematics. Their rank, defined in analogy with matrix rank, is however much more poorly understood, both from a structural and algorithmic viewpoints.