As, beginning with the famous Hofstadter's butterfly, all
numerical studies of spectral and dynamical quantities related to
quasiperiodic operators are actually performed for their rational
frequency approximants, the questions of continuity upon such
approximation are of fundamental importance. The fact that continuity
issues may be delicate is illustrated by the recently discovered
discontinuity of the Lyapunov exponent for non-analytic potentials.

I will review the subject and then focus on work in progress, joint with Avila and Sadel, where we develop a new approach to continuity, powerful enough to handle matrices of any size and leading to a number of strong consequences.

This is the first of (likely) two talks, where an almost entire proof will be presented. For understanding most of the talk knowledge of spectral theory should not be necessary and just knowing some basic harmonic analysis should suffice.

Abstract: The comparison isomorphism in p-adic Hodge theory asserts that
in some sense, the p-adic etale cohomology and the algebraic de Rham cohomology
of a smooth proper variety over a finite extension of Q_p determine each
other. We propose an alternate interpretation in which the central
object is a standard auxiliary object in p-adic Hodge theory called a
(phi, Gamma)-module, from which p-adic etale cohomology and algebraic de
Rham cohomology are functorially derived using mechanisms introduced by
Fontaine. The hope is to then enrich this object to carry additional
structures especially for varieties defined over number fields; we
illustrate this by showing how to incorporate the rational structure of
de Rham cohomology. (This depends on joint work with Chris Davis.)

In the 80s Gitik proved the following theorem: For every real $x$ and every club $D \subseteq [\omega_2]^\omega$, there are $a,b,c \in D$ such that $x \in L(a,b,c)$. An immediate corollary of Gitik's theorem is: if $W$ is a transitive $ZF^-$ model of height at least $\omega_2$ such that $W$ is missing some real, then the complement of $W$ is stationary in $[\omega_2]^\omega$ (Velickovic strengthened Gitik's Theorem to show that the complement of such a $W$ is in fact projective stationary, not just stationary). I will present Gitik's proof and, if time permits, discuss some recent applications due to Viale and me.

Talk Abstract:
The Nobel Prize-winning discovery of quasicrystals has spurred much work in aperiodic sequences and tilings. One such example is the family of one-dimensional discrete Schrödinger operators with potentials given by primitive invertible substitutions on two letters, which are a one-dimensional model of quasicrystals. We prove results about spectral properties of these operators using tools from hyperbolic dynamics.

In this talk, we study  the Gromov-Hausdorff convergence of  Ricci-flat metrics   under degenerations of Calabi-Yau manifolds. More precisely, for a family of polarized Calabi-Yau manifolds degenerating to a singular Calabi-Yau variety,  we prove that  the Gromov-Hausdorff limit of Ricci-flat kahler metrics on them is unique.