We will look at the distribution of the zeroes of the zeta
functions of Artin-Schreier covers over a fixed finite field of
characteristic $p$ as the genus grows. We will focus on two cases: the
$p$-rank zero locus and the ordinary locus.

There are beautiful and unexpected connections between algebraic
topology, number theory, and algebraic geometry, arising from the study of
the configuration space of (not necessarily distinct) points on a variety.
In particular, there is a relationship between the Dold-Thom theorem, the
analytic class number formula, and the "motivic stabilization of symmetric
powers" conjecture of Ravi Vakil and Melanie Matchett Wood. I'll discuss
several ideas and open conjectures surrounding these connections, and
describe the proof of one of these conjectures--a Hodge-theoretic
obstruction to the stabilization of symmetric powers--in the case of curves
and algebraic surfaces. Everything in the talk will be defined from
scratch, and should be quite accessible.

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.