In this talk, I will explain the notion of Hofer energy of
J-holomorphic curves in a noncompact symplectic manifold M. If M
comes from puncturing a closed symplectic manifold, we prove that the
Hofer energy can by bounded by a constant times the symplectic
energy. As an immediate consequence, we prove a version of Gromov's
monotonicity theorem with multiplicity for J-holomorphic curves.

Recently there has been considerable activity in the study of the dynamics of these groups and this work has led to interesting interactions between logic, finite combinatorics, group theory (both in the topological and algebraic context), topological dynamics, ergodic theory and representation theory. In this lecture I will give a survey of some of the main directions in this area of research.
 

I will discuss a number of related conjectures concerning the rational points of varieties (especially curves and abelian varieties) over fields with finitely generated Galois group and present some evidence from algebraic numebr theory, Diophantine geometry, and additive combinatorics in support of these conjectures.

We prove that the product of two Cantor sets of large thickness is an interval in the case when one of them contains the origin. We apply this result to the Labyrinth model of a two-dimensional quasicrystal, where the spectrum is known to be the product of two Cantor sets, and show that the spectrum becomes an interval for small values of the coupling constant. We also consider the density of states measure of the Labyrinth model, and show that it is absolutely continuous with respect the Lebesgue measure for most values of coupling constants.

Advisors:  Professor Timur Oikhberg and Lana Jitomirskaya