Notes for potential students

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 main research 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 working in arithmetic geometry to be comfortable with this material by the end of their third year. (Of course, if you are working in a different area, the core knowledge you need may be different!)

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 really enough) and they may not meet at a convenient time, or you may want to have command of the material sooner than the next scheduled course offering!

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. One aspect of this field my students and I have often explored is the rich geometry that arises from the study of arithmetic problems over function fields of curves over finite fields; I wrote lecture notes for the 2014 Arizona Winter School which give a pretty good summary of how I see this circle of ideas. I am also thinking a lot about stability phenomena in topology and representation theory (see e.g. "FI-modules over Noetherian rings,", a joint paper with my student Rohit Nagpal) and about algebro-geometric methods in combinatorics (see e.g. this paper with Daniel Erman.) I definitely do not require that you work on problems directly related to my own research.

Where will I get my thesis problem? The very 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. And the majority of my students have written theses on problems which originated, at least in part, from questions I suggested.

What is the schedule? 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).

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