Over 30 lectures in the winter of 2012 I taught an introduction to
$p$-adic Hodge theory to Caltech graduate students. The notes start
with Sen theory treating in detail the decompletion procedure. The
majority of these notes are concerned with admissible $p$-adic Galois
representations and Fontaine's construction of
$B_{\operatorname{dR}}$, $B_{\operatorname{cris}}$, and
$B_{\operatorname{st}}$. When possible, I simplified proofs by
going instead to a similar but simpler setting, e.g.,
$B_{\operatorname{max}}$ instead of $B_{\operatorname{cris}}$. The
last section is concerned with the basic computations of Bloch and
Kato on ordinary Galois representations.
After reviewing the main results of global class field theory,
including more advanced topics such as global duality and Selmer
groups, the notes are concerned with disparate topics that use global
class field theory:
Mordell-Weil for elliptic curves. Rather than reproducing the
standard proof found, e.g., in Silverman, I explain how the elliptic
curve Selmer group is the Selmer group of a Selmer
system. Mordell-Weil is then an immediate corollary.
Grunwald-Wang and the existence of global characters with
prescribed behavior.
Tate's theorem on lifting projective Galois representations.
Iwasawa theory for $\mathbb{Z}_p$-extensions and the
$p$-adic valuation of the class number in the $p$-cyclotomic
tower. On its own, this is a semester-long course, I concentrated
on classifying $\mathbb{Z}_p$-extensions and Leopoldt and modules
over the Iwasawa algebra.
Hecke theory for characters, an overview of Tate's thesis.
Deligne's construction of local $\varepsilon$-factors for Galois representations.
These notes cover the main topics of a first course in graduate
algebraic number theory, with two differences: I used adelic language
to explain how Minkowski's geometry of numbers becomes a simple
topological statement, and I explain how to compute special values of
$L$-functions of Dirichlet characters. In my previous notes I did not
include the adelic language, but I included a peculiar integral
computed using $L$-functions. These notes are best consulted together
with their accompanying problem sets.
These notes include the main topics of a basic graduate algebra course:
finite groups, the Sylow theorems, finite group actions, rings,
modules, fields, Galois theory. In addition, the first set of notes
include injective and projective limits of groups, while the second
set of notes includes topics on homological algebra (derived functors,
injective and projective modules, limits), and infinite Galois groups.
The undergraduate workshop consisted of three coordinated lecture series: Gábor
Székelyhidi on Riemann surfaces, Claudiu Raicu on The Riemann-Roch
formula, and myself on Modular forms.