It is well known that there exist several differentiable or topological obstructions to compact manifolds admitting metrics of positive scalar curvature. On the other hand, the family of manifolds with positive scalar curvature is quite large since any finite connected sum of them is still a manifold admitting a metric of positive scalar curvature. This talk is concerned with the classification question to this family.
         The classical uniformization theorem implies that a two-dimensional compact manifold with positive scalar curvature is diffeomorphic to the sphere or the real projective space. The combination of works of Scheon-Yau and Perelman gives a complete classification to compact three-dimensional manifolds with positive scalar curvature. In this talk we will discuss how to extend Schoen-Yau-Perelman's classification to four-dimension. This is based on the joint works with Bing-Long Chen and Siu-Hung Tang.

We present fast frequency- and time-domain spectrally accurate solvers for Partial Differential Equations that address some of the main difficulties associated with simulation of realistic engineering systems. Based on a novel Fourier-Continuation (FC) method for the resolution of the Gibbs phenomenon and fast high-order methods for evaluation of integral operators, these methodologies give rise to fast and highly accurate frequency- and time-domain solvers for PDEs on general three-dimensional spatial domains. Our fast integral algorithms can solve, with high-order accuracy, problems of electromagnetic and acoustic scattering for complex three-dimensional geometries; our FC-based differential solvers for time-dependent PDEs, in turn, give rise to essentially spectral time evolution, free of pollution or dispersion errors, for general PDEs. A variety of applications to linear and nonlinear PDE problems demonstrate the significant improvements the new algorithms provide over the accuracy and speed resulting from other approaches.
 

It is a classical result, by Dyson, that the behavior of the eigenvalues of a random unitary matrix following uniform measure tend, when the dimension goes to infinity, after a suitable scaling, to a random set of points, called adeterminantal sine-kernel process. By defining the model in all dimensions on a single probability space, we are able to show that the convergence stated above can occur almost surely. Moreover, in an article with K. Maples and A. Nikeghbali, we interpret the limiting point process as the spectrum of a random operator.

We consider the contact process on a random graph chosen with a fixed degree, power law distribution, according to a model proposed by Newman, Strogatz and Watts (2001). We follow the work of Chatterjee and Durrett (2009) who showed that for arbitrarily small infection parameter λ