Jack Thorne.
Etale Cohomology of Number Fields.
Etale cohomology was developed by Grothendieck, with assistance from M. Artin, in the years 1963-64. The motivation for this was the well-known strategy for the proof of the Weil conjectures. On the other hand, it is easy to see that one recovers Galois cohomology as a special case of etale cohomology over the spectrum of a field, and it is natural to ask what happens when ones works over the spectrum of the ring of integers in a number field. Artin and Verdier were the first to do this, and proved a duality theorem of Poincare type. I will spend some time recalling the basics of etale cohomology, make some computations, and sketch a proof and some applications of the duality theorem. I will be following Mazur's paper "Notes on the etale cohomology of number fields".
The alcove seminar's name originates from the traditional venue for it's meetings: the small alcoves spotting the halls of the math department. The seminar is aimed at being more approachable than the department number theory seminar.