Carl Erickson.
Some Obstructions to Rational Points on Varieties.
These talks will attempt to highlight portions of Bjorn Poonen's course at MIT last semester, "Rational Points on Varieties." His very complete course notes can be found here. We will give an indication of the state of knowledge of rational points on varieties to start. From then on we will focus on the central theme of these talks, which is determining whether a variety has any rational point whatsoever. After providing a few basic algebraic geometric ideas with bearing on rational points, we will introduce torsors and the accompanying cohomology theories in order to work toward explaining the descent and Brauer-Manin obstructions to the local-global principle. These are the only known obstructions. To conclude, we hope to mention on one hand the situations where these obstructions are conjectured to give all exceptions to the local-global principle, and then on the other hand give an example of the insufficiency of these obstructions.
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.