Notes for graduate students considering working with me
Background
I am currently happy to mentor students who are interested in arithmetic statistics, including of functions fields, and also to mentor students who are interested in the probability theory of random groups, either in the pure probability theory, or in the application of these ideas to other fields (especially including topology and combinatorics).
If you are interested in arithmetic statistics, you will need to have a strong foundation in Algebraic Number Theory (e.g. see Milne's course notes) and Class Field Theory (e.g. see Milne's notes).
Depending on your direction of interest, you will need to add probably at least two of the following
- Algebraic Geometry (varieties and schemes, e.g. Harris \emph{Algebraic Geometry: A First Course}, Chapters 1-4, Appendix C of Hartshorne, \emph{Geometry of Schemes})
- Algebraic Topology (e.g. Hatcher)
- Representation Theory (e.g. of finite groups, Serre \emph{Linear Representations of Finite Groups} and Lie algebras in Fulton and Harris)
- Analytic Number Theory (e.g. Davenport's \emph{Multiplicative Number Theory})
- Probability (e.g., Durrett, Probability Theory and Examples)
- Etale cohomology (e.g. Milne's notes)
- Elliptic curves (e.g. Silverman, \emph{The Arithmetic of Elliptic Curves})
If you are interested in the probability theory of random groups, you should have a strong foundation in probability (Durrett, Probability Theory and Examples), algebra (at the level of the Harvard qual), and probably some further topic of interest for developing applications, such as topology, number theory, combinatorics, algebraic geometry, or representation theory.
You do not need to have all of this background before we start talking or meeting. The sooner we start talking, the better.
Deciding on an advisor
My usual plan is that I should meet with a student regularly for a semester while they are reading some papers or working on a small project, and after a semester we can have a discussion about whether it is a good advisor/student fit.