It is proved that if a compact manifold admits a smooth action by a compact, connected, non-abelian Lie group, then it admits a metric of positive scalar curvature. This result is used to prove that if ^<i>n</i> is an exotic<i>n</i>-sphere which does not bound a spin manifold, then the only possible compact connected transformation groups of ^<i>n</i> are tori of dimension [(<i>n</i>+1)/2].

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We consider $N\times N$ Hermitian or symmetric random matrices with independent entries. The distribution of the $(i,j)$-th matrix element is given by a probability measure $\nu_{ij}$ whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution $\nu_{ij}$ coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector-eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.

No abstract uploaded!

No abstract uploaded!