i will report on my recent joint work with greither on the galois module structure of $\ell$-adic realizations of picard $1$-motives in positive characteristic and their iwasawa theoretic analogues in characteristic 0. in the process, i will state and sketch the proof of an equivariant main conjecture in iwasawa theory. finally, i will discuss applications of the equivariant main conjecture ranging from proofs of refinements of the brumer-stark and coates-sinnott conjectures to proofs of particular cases of the equivariant tamagawa number conjecture of bloch-kato and the non-abelian main conjecture of coates-fukaya-kato-sujatha-venjakob. this is a two hour lecture, the first half emphasizing the more geometric aspects of our work, while the second will focus on the number theoretic aspects and applications. talk time runs until 4:00 pm.