In 1953 Roth proved that any positive density subset of the integers contains a non-trivial three term arithmetic progression. I will present a recent quantitative improvement for this theorem, give an overview of the main ideas of the proof, and discuss its relation to other recent work in the area. I will also discuss some closely related problems. 

We discuss a new method to bound the number of primes in certain very thin sets. The sets $S$ under consideration have the property that if $p\in S$ and $q$ is prime with $q|(p-1)$, then $q\in S$. For each prime $p$, only 1 or 2 residue classes modulo $p$ are omitted, and thus the traditional small sieve furnishes only the bound $O(x/\log^2 x)$ (at best) for the counting function of $S$. Using a different strategy, one related to the theory of prime chains and Pratt trees, we prove that either $S$
contains all primes or $\# \{p\in S : p\le x \} = O(x^{1-c})$ for some positive $c$. Such sets arise, for example, in work on Carmichael's conjecture for Euler's function.

In this talk, we will discuss the dimension of $J_0(N)[m]$ at an Eisenstein prime m for
square-free level N. We will also study the structure of $J_0(N)[m]$ as a Galois module.
This work generalizes Mazur’s work on Eisenstein ideals of prime level to the case of
arbitrary square-free level up to small exceptional cases.

Let f be an elliptic modular newform of weight at least 2. The 
problem of the automorphy of the symmetric power L-functions of f is a 
key example of Langlands' functoriality conjectures. Recently, the 
potential automorphy of these L-functions has been established, using 
automorphy lifting techniques, and leading to a proof of the Sato-Tate 
conjecture. I will discuss a new approach to the automorphy of these 
L-functions that shows the existence of Sym^m f for m = 1,...,8.