There is a classical theorem of Iwasawa which concerns certain modules X for the formal power series ring Λ = Zp[[T]] in one variable.  Here p is a prime and Zp is the ring of p-adic integers.  Iwasawa's theorem asserts that X has no nonzero, finite Λ-submodules. We will begin by describing the modules X which occur in Iwasawa's theorem and explaining  how the theorem is connected with the title of my talk. Then we will describe generalizations of this theorem for  certain Λ-modules (the so-called "Selmer groups")  which arise naturally in Iwasawa theory.  The ring Λ can be a formal power series ring over Zp in any number of variables, or even a non-commutative analogue of such a ring. 

The overconvergent modular symbols of Stevens provide a natural framework for computing p-adic L-functions of newforms, but the modular symbols (and p-adic L-functions) attached to ordinary Eisenstein series are essentially trivial. Working with a larger space of pseudo-distributions, we construct non-trivial Eisenstein symbols and compute their p-adic L-functions. As a corollary, we compute the p-adic L-function of the "evil twin" Eisenstein series of critical slope. If time permits, I'll discuss work in progress on computing the symmetric square p-adic L-function at Eisenstein points on the eigencurve, as well as applications.

The classic Linear Preserver Problem asks to determine, for a polynomial function f on a vector space V, the linear transformations g of V such that fg = f. In case f is invariant under a simple algebraic group G acting irreducibly on V, we prove that the subgroup of GL(V) stabilizing f often has identity component G and we give applications realizing various groups, including the largest exceptional group E8, as automorphism groups of polynomials and algebras. We show that starting with a simple group G and an irreducible representation V, one can almost always find an f whose stabilizer has identity component G and that no such f exists in the short list of excluded cases. The main results are new even in the special case where the field is the complex numbers, and have implications for Hasse principles for polynomials over number fields.  This talk is about joint work with Bob Guralnick. 

See the abstract here.