I’ve been thinking about what I’d like to write about on this blog, and I have some ideas to keep me busy for the next little while. Yesterday I introduced the notion of symmetry and mentioned the notion of an abstract group. Over the next few days I’d like to discuss a related topic that is close to my heart: the symmetries of numbers. For this I’ll need to introduce groups and number fields, and I’ll try to explain some aspects of Galois theory in an accessible fashion. I’d like to follow this up sometime in the future with a discussion of how the Monster group can be realized as a Galois group over the rationals.

The Monster group will have to wait for awhile, as Vladimir Berkovich is visiting Montreal for the next two weeks. He’s giving a series of 3 lectures, to be followed by an open discussion session. The first lecture takes place on Thursday November 5 at Concordia University, as part of the Quebec-Vermont Number Theory Seminar. An abstract hasn’t been posted yet, but I believe this will be a more advanced research level talk on p-adic integration of one-forms. On the following Thursday November 12, Berkovich will give two introductory lectures on his analytic spaces, assuming background in rigid analytic geometry and Raynaud’s approach to p-adic geometry via formal schemes. Finally, on Friday November 13, there will be an open discussion session on Berkovich spaces. As I’d eventually like to understand how these spaces arise in Harris-Taylor’s work on the local Langlands conjecture, it seems like a good time to learn some more about Berkovich’s work. So expect some posts on this as well!

Oh, and if you’ve never heard of Berkovich or his work before, you should definitely check this out from his personal webpage. It’s taken from the AMS volume on the 2007 Arizona Winter School on p-adic Geometry, and describes Berkovich’s discovery of his analytic spaces. Some interesting personal details are included about the discrimination that Berkovich faced in Russia at the time. In particular, despite his brilliance as a mathematician, as a Jew he had a difficult time getting an academic position in mathematics. He had to settle for a job as a computer programmer, and learned to work on mathematics during odd free hours of the working day, or else very early in the morning (his evenings were devoted to his family!). These personal reflections make for an entertaining read for mathematicians and nonmathematicians alike, although you may wish to skim through some of the mathematics.