Oscar Villareal will lead a discussion on the NTRU Prime paper.

In this talk, we will discuss Furstenberg correspondence principle, which connects ergodic theory and combinatorial number theory. 

A "critical" metric on a manifold is a metric which is critical for some natural geometric variational problem. Some important examples of critical metrics are Einstein metrics and extremal Kahler metrics, and such metrics typically come in families. I will discuss some aspects of the local theory of moduli spaces of critical metrics, and present some compactness results for critical metrics which say that, under certain geometric assumptions, a sequence of critical metrics has a subsequence which converges, in the Gromov-Hausdorff sense, to a singular space with orbifold singularities. I will also discuss some results regarding the reverse problem of desingularizing critical orbifolds to produce new examples of critical metrics on smooth manifolds.
 

The topic of this talk will be understanding the p-adic slopes of modular forms. Recently, Bergdall and Pollack, based on computer calculations, raised a very interesting conjecture on the slopes of overconvergent modular forms, which predicts that the Newton polygons of the characteristic power series of U_p are the same as the Newton polygons of another explicit characteristic power series, which they call ghost series. This conjecture would imply many well-known conjectures regarding slopes of modular forms, like Gouvea's conjecture, Gouvea-Mazur conjecture, and etc. The goal of our joint project is to prove this conjecture under some mild hypothesis, and to explore some further application.  I will report on the progress so far.