In the first part of the talk, using ideas from Ricci flow, we get a Li-Yau type gradient estimate for positive solutions of heat equation on Riemmannian manifolds with $Ricci(M)\ge -k$, $k\in \mathbb R$.
In the second part, we establish a Perelman type Li-Yau-Hamilton differential Harnack inequality for heat kernels on manifolds with $Ricci(M)\ge -k$. And we obtain various monotonicity formulas of entropy.
Parabolic Harnack inequalities were proved by Moser for linear equation with bounded and measurable coefficients. In the case of the parabolic p-Laplacean such kind of estimates cannot hold (as proved by Trudinger). In the nineties DiBenedetto introduced the so called intrinsic Harnack inequalities for the protype equation. His original proof requiries the maximum principle and the existence of suitable subsolutions. Therefore the proof for general equations (with bounded and measurable coefficient) was missing. In some recent papers, in collaboration with DiBendetto and Gianazza, we proved intrinsic Harnack inequalities for general degenerate and singular operators. We show, via suitable counterexamples, that such estimates are sharp. Moreover we proved that when p is approaching to 2, our estimates tend to the classical Moser estimates.
I will present a novel multiscale clustering algorithm inspired by algebraic multigrid techniques. Our method begins with assembling
data points according to local similarities. It uses an aggregation process to obtain reliable scale-dependent global properties, which arise from the local similarities. As the aggregation process proceeds, these global properties influence the formation of coherent clusters. The
global features that can be utilized are for example density, shape, intrinsic dimensionality and orientation. The last three features are a
part of the manifold identification process which is performed in parallel to the clustering process. The algorithm detects clusters that
are distinguished by their multiscale nature, separates between clusters with different densities, and identifies and resolves intersections between clusters. The algorithm is tested on synthetic and real data sets, its running time complexity is linear in the size of the data set.
The Primitive Equations are a fundamental model describing large scale oceanic and atmospheric processes. They are derived from the fully compressible Navier-Stokes equations on a combined basis of scale analysis and meteorological data. While an extensive body of mathematical literature exists in the study of these systems, very little is known in the stochastic setting. In this talk we discuss recent joint work with M. Ziane concerning existence and uniqueness of solutions for the 2-D equations in the presence of multiplicative noise terms.