We will present two different ways to generically add a club subset of a successor cardinal, under some GCH.  The first one is designed to destroy a given stationary set, and we show that it also forces diamond.  The second adds a club with "small" conditions and destroys saturated ideals.  We will discuss the open problem of whether this can be done without any cardinal arithmetic assumptions.

A theorem of Eichler and Shimura says that the space of cusp forms with complex coefficients appears as a direct summand of the cohomology of the compactified modular curve. Ohta has proven an analog of this theorem for the space of ordinary p-adic cusp forms with integral coefficients. Ohta's result has important applications in the Iwasawa theory of cyclotomic fields.

We discuss a new proof of Ohta's result using the geometry of Hecke correspondences at places of bad reduction.

The study of isoperimetric inequalities has a long history,
it's humble beginnings in Ancient Greek mathematics belying a deep and
rich theory. A major tool in the study of the isoperimetric profile is
the Calculus of Variations. Variational arguments lead to weak
differential inequalities for the isoperimetric profile, which allows
analytical tools such as the maximum principle to be employed. Of
central importance here is the connection with curvature, which is
intimately connected with the topology of isoperimetric regions. I
will survey some of the results in this direction, paying particular
attention to the interplay of the isoperimetric profile and curvature
flows which is the focus of my current research.

For any irregular prime p, one has a Hida family of cuspidal eigenforms of level 1 whose residual Galois representations are all reducible. This family has already played a starring role in Wiles’ proof of Iwasawa’s main conjecture for totally real fields. In this talk, we instead focus on the Iwasawa theory of these modular forms in their own right. We will discuss new phenomena that occur in this residually reducible case including the fact that analytic mu-invariants are unbounded in this family and directly related to the p-adic zeta-function. This is a joint work with Joel Bellaiche.