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!)

- Algebraic number theory at the level of one of the standard
texts; for instance, the first two chapters of Neukirch's
*Algebraic Number Theory*. - Silverman,
*The Arithmetic of Elliptic Curves*. - Hartshorne,
*Algebraic Geometry*, the first four chapters. In this case, "comfortable" means "I have done all or most of the exercises." - Galois cohomology at the level of Serre's book; ideally, some etale cohomology at the level of Milne's book.

**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).

Back to Jordan Ellenberg's home page

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