The space of (Kahler) metrics on a projective algebraic variety can be given a natural infinite dimensional Riemannian structure.  This leads to the notion of a geodesic in the space of metrics.  I will discuss a recent result on the optimal regularity of these geodesics and how this relates to nonlinear PDEs and canonical metrics.  This is joint work with J. Chu and V. Tosatti.

We give a new and short proof of spectral localization for the 1-d Anderson model with any disorder. The original proof was given by Carmona-Klein-Martinelli in 1987, based on multi-scale analysis. Our proof is based on the large deviation estimates and positivity and subharmonicity of the Lyapunov exponent. We also show how to improve the estimates to get a uniform and quantitative version that allows us to get the exponential dynamical localization.  It is joint work with S. Jitomirskaya. Complete details of the spectral localization proof will be presented during the talk. 

We will start from the motivation of the tropical geometry. Then
we will explain how to use Lagrangian Floer theory to establish the
correspondence between the weighted counting of tropical curves to the
counting of holomorphic discs in K3 surfaces. In particular, the result
provides the existence of new holomorphic discs which do not come easily
from direct gluing argument.

In this workshop we will discuss techniques to facilitate group work, and make discussion sessions more active. We will also introduce tasks aimed at actively involve students in various phases of the learning process (e.g., introduce/explore/review a topic, learn the steps to solve a particular problem or the lay out of a proof).

We describe a number of related questions at the interface of set theory and homology theory, centering on (1) the additivity of strong homology, and (2) the cohomology of the ordinals. In the first, the question is, at heart: To how general a category of topological spaces may classical homology theory be continuously extended? And in the tension between various potential senses of continuity lie a number of delicate set-theoretic questions. These questions led to the consideration of the Cech cohomology of the ordinals; the surprise was that this is a meaningful thing to consider at all. It very much is, describing or suggesting at once (i) distinctive combinatorial principles associated to the nth infinite cardinal, for each n, holding in ZFC, (ii) rich connections between cofinality and dimension, and (iii) higher-dimensional extensions of the method of minimal walks.