Notes for potential students

(idea for this page copied from Kiran Kedlaya's excellent page on the same topic...)

If you are thinking about whether I would be a good Ph.D. advisor for you, here is some information that might be useful.

What math you should know. My area is arithmetic geometry, which has the somewhat vexing feature of offering many problems which are easy to state, but which require quite a bit of machinery to address meaningfully. Precisely what machinery you will need depends on what problem you work on, but here is a list of books with which anyone planning to go into this area should be comfortable. You don't have to know everything on this list when you start, but I would expect any student of mine to have mastered this material by the third year.

Of course, this does not come close to exhausting the subjects that will be useful for research in arithmetic geometry. It's hard to imagine you would not find good use for a basic knowledge of: analytic number theory, the representation theory of finite groups, abelian varieties, automorphic forms, Galois representations, moduli of curves and the moduli space point of view in general, Hodge theory, algebraic topology….

What's the best way to learn all this stuff? For some people, reading books and doing exercises is the best way. For others (perhaps for most) it's better to make it a bit more social: form a group of grad students, read a section a week, writing down all your questions and working out examples to aid your reading. Then convene weekly and trade questions and comments. Some of these topics are covered by grad courses at Wisconsin, but these do not always cover the right amount of material (e.g. the amount of algebraic geometry you can learn in a one-semester course is not enough) and they may not meet at a convenient time.

What kind of math are you thinking about? I have fairly broad interests, but most things I think about involve, in one way or another, the problem of studying rational points on algebraic varieties over global fields. There are many points of view on this part of mathematics: this set of lecture notes and open problems exposes one of them. The lecture notes from the Rings of Low Rank conference at Leiden represents another. I also like modular forms and Galois representations, and am happy to supervise students in those areas. I definitely do not require that you work on problems directly related to my own research.

Where will I get my thesis problem? The best way to get a thesis problem is to find one yourself. The best way to find one yourself is to expose yourself to lots of problems; for instance, by coming to number theory and algebraic geometry seminars every week, by looking over new papers posted on the arXiv, and by going to conferences in number theory and arithmetic geometry. Make a resolution that whenever you go to a seminar talk or look at a paper, you are going to formulate some question about the topic of the paper. It might turn out to be trivial, might turn out to be impossible, but it will get you in the habit of thinking up questions; and it will give you something to talk about with the speaker after the talk.

That said, I am happy to suggest potential problems or to refine and tweak ones that you bring to me.

What is the schedule? As you know, UW does not guarantee funding beyond the fifth year. The schedule is as follows: First year, take courses. Second year, learn number theory and algebraic geometry and work on a "toy problem" which will hopefully lead to a publication in addition to your thesis. Second or third year, take your specialty exam, explaining what you've done so far and fielding questions about the "essential" topics as listed at the top of this page. Third and/or fourth year, solve your thesis problem. Fourth or fifth year, apply for jobs, write up thesis, get started on your next big project!

What else should I read? Guillermo Mantilla-Soler suggested that I add some recommended books to this page. Some books that I find I use all the time, besides those already mentioned above, are: Serre's Local Fields and Linear Representations of Finite Groups and Oeuvres; Neron Models (Bosch, Lutkebohmert, Raynaud); Arithmetic Geometry (Cornell, Silverman); Cohomology of Number Fields and Algebraic Number Theory (Neukirch); Modular Forms and Fermat's Last Theorem (Cornell, Stevens, Silverman); Diophantine Geometry (Hindry, Silverman). Snap these up when you see them on sale!

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Jordan Ellenberg * ellenber@math.wisc.edu * revised 21 Dec 2006