Notes for prospective PhD advisees

Besides teaching and research, one of the principal activities of my profession is the direction of PhD dissertations. This page is meant to describe my process for advising PhD students, for the benefit of those considering doing a thesis under my supervision. This description is of course from my point of view; you may wish to consult my former or present students from an alternate perspective. Although some of this advice is specific to me, quite a lot of it may be applicable to graduate students in mathematics more broadly.

Last modified Monday, 27-Aug-2018 18:18:47 EDT.

My areas of research

I'll start by giving a general description of my mathematical interests. For more specific discussion of research topics, I have a separate page.

The greater part of my research is in the field of arithmetic algebraic geometry. This subject lies at the boundary between algebraic number theory and algebraic geometry; one is naturally led to this boundary from many traditional questions of number theory. Besides questions about algebraic varieties over number fields, one often considers varieties over finite fields; this leads to some profitable analogies, e.g., between the ring of integers and the ring of univariate polynomials over a finite field.

Within arithmetic algebraic geometry, I am particularly interested in problems of p-adic analysis. This subject is a variant of analysis over the real or complex numbers, which allows ideas from this subject (which might be considered continuous mathematics) to be applied to the study of discrete objects (such as the integers).

By its nature, number theory is a subject largely driven by explicit computations. These were once made by hand, but nowadays the presence of computers opens up many more avenues for numerical investigation. However, the deployment of computers in number theory is not automatic; a rich field of computational number theory exists at this interface. The flow of ideas between theory and computations proceeds in both directions: computations lead to numerical evidence to devise or support conjectures, while theoretical results often lead to unexpected new computational methods.

That all having been said, my interests are fairly diverse, and often bleed over into adjacent areas of mathematics, such as classical algebraic geometry, combinatorics, theoretical computer science...

Preparation

The appropriate level of preparation for work in this areas depends quite a bit on one's specific tastes. However, I generally expect my students to have good familarity with algebraic number theory (including local fields, class field theory, and Galois cohomology), and with algebraic geometry both in the language of varieties (as in the books of Mumford or Shafarevich) and schemes (as in the book of Hartshorne, or for the dedicated, the first part of Grothendieck's EGA).

In practice, I will typically propose background reading for starting students, based on their prior preparation and on the problem(s) I have in mind. One important point to keep in mind is that unlike at earlier stages in one's education, in graduate school and beyond one very often learns mathematics "in reverse". That is, it is often necessary to work through a particular piece of mathematics before one is familiar with its logical prerequisites; the switch from forward to reverse study of mathematics is quite a difficult transition (at least according to my own experience as a student).

How to find a problem

Matching students with problems is a complicated and difficult art. Some advisors take a completely hands-off approach; for instance, Hendrik Lenstra claims that he can no more suggest to his students what problems they should work on than who they should marry.

In my experience, relatively few students manage this without any assistance, so I tend to take a somewhat more interventionist approach. I tend to focus on a small number of areas, which I discuss on my page of questions of interest. I am happy to discuss these questions to see whether one of these may lead to a suitable problem.

On the other hand, I also strongly approach students to discuss with other mathematicians about possibly interesting problems. I tend to know a little bit about many topics (and am willing to learn more should the situation warrant), so it is quite practical to work with me on a problem suggested by someone else. Indeed, several of my students so far have done exactly this.

It is generally beneficial for students to have more than one project in progress at once. It is okay if not everything one is doing in graduate school ends up in the thesis, as long as it all eventually gets disseminated (see below).

Supervision

It is the joint responsibility of advisor and student to meet often enough to ensure that adequate progress is being made. I generally try to meet with students on a regular basis (say weekly) to the extent possible. That said, I travel frequently, so this type of regular schedule tends to get perturbed in practice.

Dissemination

It is not absolutely necessary for students to publish their work before completing their dissertation. At this stage in one's career, what is more important than publication is dissemination, i.e., making sure that one's work comes to the attention of those to whom it will be of greatest interest. Formal publication is one medium for dissemination; others include posting a preprint to one's web site or to a public server such as arXiv, (e)mailing a paper to appropriate mathematicians, or even simply discussing one's work with others.

The importance of dissemination cannot be understated, particularly for those planning to pursue a career in academia. Advancement at all stages is driven largely by letters of recommendation, so it is important to keep colleagues informed of one's progress. For graduate students planning to continue to a postdoc, normally three (or more) letters of recommendation are needed. One will be from the thesis advisor; ordinarily a second will be from someone else at the student's institution, and a third will be from an outside mathematician familiar with the area of reseach. It is important to develop these contacts before the time comes to ask for the letter, as we are all very busy and cannot be counted on to learn about and comment on a new topic in a couple of weeks.

This all being said, I should also caution against premature dissemination. Because mathematics is a domain in which something resembling absolute truth can be established, mathematicians have a far lower tolerance for error than other scientists. Those who habitually overstate their results run the risk of becoming mathematical pariahs, for whom no one will invest the time to understand their work. Avoid.

It should also be noted that even in the arXiv era, publication remains important for career advancement beyond graduate school. The publication process includes the important component of peer review. Acceptance of a paper for publication in a journal or conference, while not a guarantee of absolute correctness, is an important mark of validation. (Which journal or conference it happens to be does carry some weight, but early in one's career it is probably best not to fixate on this too closely.)

Rate of progress

The typical length of a PhD program varies among institutions. At UCSD, my general expectation is that students working with me should complete a thesis within five years of entering the program.

Regardless of the institution, it is a good idea for students to start "shopping around" for an advisor early on, even before settling on an area of specialization. In fact, often the choice of an advisor will dictate the choice of area, rather than vice versa.

It is also an extremely good idea to make sure that one acquires adequate teaching experience, since most job applications require a letter of reference concerning teaching. Such a letter may be written, e.g., by an instructor for whom you worked as a teaching assistant, or by a department member who supervises teaching in some fashion. It is a good idea to solicit such a letter shortly after the teaching in question takes place, rather than waiting until one's final year of graduate school.

Those with outside funding might be able to postpone acquiring this experience until the year of graduation, but this is strongly discouraged, as your second teaching experience is likely to turn out better than the first. On the other hand, those with extensive teaching expertise may benefit from reductions of their teaching burden. One's advisor can sometimes provide a research assistantship from grant funds; there may also be research opportunities in government or industry that may be relevant. Such opportunities are often not directly related to one's thesis problem, but may at least be in one's research area, and may even lead to interesting side projects, and occasionally even to unexpected career opportunities.