Calculus means the differentiation and integration of functions and differential forms, and leads naturally to the notion of de Rham cohomology. Classical Hodge theory provides harmonic representatives for de Rham cohomology classes. Its more recent nonabelian version associates a Higgs bundle to a representation $
ho$ of the fundamental group and Higgs classes to $
ho$-twisted cohomology classes. Calculus and de Rham cohomology make sense algebraically and in any characteristic, but Hodge theory is profoundly analytic. Nevertheless I will describe recent attempts to construct an analog of nonabelian Hodge theory in characteristic $p$ (joint work in progress with V. Vologodsky).

It is known that a simply connected manifold with positive isotropic curvature (PIC) is homeomorphic to a sphere. The observation that the product metric on $S^1 imes S^n$ has PIC and the fact that the class of manifolds with PIC is closed under connected sums led to the conjecture that the fundamental group of manifolds with PIC is almost free. Significant progress on this conjecture has recently been made by Ailana Fraser. In this talk I will describe the notion of almost PIC (which still implies positive scalar curvature) and I will indicate why this class of manifolds is closed under surgery over a circle. In particular, there is no restriction on the fundamental group for manifolds with almost PIC. By considering higher dimensional surgeries, there is even reason to believe that the class of simply connected manifolds with almost PIC coincides with that of positive scalar curvature. This is joint work in progress with Ingi Petursson.

testing

We consider a symmetric $alpha$-stable model for heavy tailed time series, which allows for some dependence structure (or memory) in the data. In this context, an estimator for the tail index is $alpha$ is presented, which has a rapid rate of convergence -in particular $Op(n^{-1/2})$ -which is robust under intermediate memory. There is no need for blocking or tuning parameters for this estimator. Small sample results and full asymptotics are provided in this paper, and simulation studies on various $alpha$-stable data sets are given as well.