Abstract:  For the second  phase transition line $\lambda=e^{\beta}$ of  almost Mathieu operator, we prove that for dense $\alpha$, the operator has purely singular continuous spectrum for every phases, and  for dense $\alpha$, the operator has pure point spectrum for almost every phases. This is joint work with Artur Avila and Svetlana Jitomirskaya. 

We will talk about a paper by A. Moreira and J.C. Yoccoz, where they proved a conjecture by Palis according to which the arithmetic sums of generic pairs of regular Cantor sets on the line either has zero Lebesgue measure or contains an interval.

We continue our discussion of perfect and scattered subsets in the generalized Cantor space. We focus in particular this week on constructing certain types of trees in 2^{<\kappa} for uncountable \kappa which exhibit fundamentally different behavior than trees in 2^{<\omega} can, from the perspective of adding branches, cardinal dichotomies, etc. We also generalize the games previously discussed, and introduce alternative notions of \kappa-perfect and \kappa-scattered.

The Almgren-Pitts min-max theory is a Morse theoretical
type variational theory aiming at constructing unstable minimal
surfaces in a closed Riemannian manifold. In this talk, we will
survey recent progress along this direction. First, we will discuss
the understanding of the geometry of the classical Almgren-Pitts
min-max minimal surface with a focus on the Morse index problem.
Second, we will give an application of our results to quantitative
topology and metric geometry. Next, we will introduce the study of
the Morse indices for more general min-max minimal surfaces arising
from multi-parameter min-max constructions. Finally, we will
introduce a new min-max theory in the Gaussian probability space and
its application to the entropy conjecture in mean curvature flow.

We prove interior H ̈older estimates for the spatial gradient of vis- cosity solutions to the parabolic homogeneous p-Laplacian equation

ut = |∇u|2−pdiv(|∇u|p−2∇u),

where 1 < p < ∞. This equation arises from tug-of-war-like stochastic games with noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.