This page is meant to provide info for students from the undergraduate to Ph.D. level on working with me on research projects. It is very much under construction, so don't hesitate to email (ben.webster@uwaterloo.ca) if you have any questions. I think it will be helpful to write it in an FAQ format.
What kind of research do you do?
It's something of a mix. I started in representation theory of Lie algebras and related structures, but that topic touches a lot of others: algebraic geometry, mathematical physics, combinatorics, topology. I tend to look for problems that can be attacked using a mix of these disciplines and one can play these different approaches off of each other. So, the good news for a student is that I can try to suggest problems that match a lot of different interests. The bad news is that I will sometimes say that in order to understand why I think a problem in one area is interesting, you have to think about a quite different one.
That was kind of vague. Can you be bit more specific?
My principal focus at the moment is understanding the representation theory and geometry of Coulomb branches; these are algebras that appeared in quantum field theory, but have purely mathematical (in fact, algebraic) definition. These include some well-known algebras from representation theory, such as universal enveloping algebras of gl_n, and Cherednik algebras. Seeing these as Coulomb branches sheds new light on their representation theory, and understanding the consequences for these algebras is a long term project with a lot of nooks and crannies; you can see this from my recent papers on the arXiv. I'm also working on other projects which are very tangentially related in homological knot invariants and other aspects of categorification, and trying to better understand quantum field theory in order to understand the larger context around all these topics.
Sounds cool. Are you accepting new students?
I am open to accepting new Masters or Ph.D. students in Fall 2024; it is now too late to apply for Fall 2023, and the Waterloo MMath and PhD programs typically don't admit students in Winter or Spring (though if you really think we should make an exception for you, you can email me about it).
So, how do I make that happen?
If you are an undergraduate at UW and have Canadian citizenship or PR, it's generally quite easy to arrange an USRA term (which can be a coop); if you're a visa student or at another university, that's more complicated, but we can discuss it.
If you're finishing a BMath or equivalent, then you should apply to the MMath program in Pure Math at UW. Probably you should email me beforehand; generally the commitee makes a point of sharing relevant applications, but things can be missed. You can also consider the PSI Masters at Perimeter, but be warned, this is a physics program, if a very theoretical one. Note that unlike in the US or UK, in Canada, a masters is required for starting a PhD program (and some UK degrees labeled Masters might still not be enough). For people familiar with graduate school in the US, you should think of the Waterloo MMath as comparable to the first year in a US Ph.D. program (it's often more advanced, since the standard first year curriculum at most US schools is part of our undergraduate curriculum); you do need to be separately admitted to the Ph.D. program, but this is not a much more serious barrier than not getting kicked out of most US programs (and strong candidates can get offers that include Ph.D. admission if certain criteria are met).
If you're already in the MMath program at UW, you should knock on my door. It will help a lot if we do a master's project together, and some conversations earlier in the year will help us figure out if that will work.
If you have or are finishing a Master's in Math from another university, then you should apply to the PhD program in Pure Math at UW. As with the MMath, the system here is a bit different than most places in the US (though familiar to Europeans): PhD students are admitted to work with a specific advisor from day one, and I will likely only consider students whose Masters background matches very well with the research I would suggest in the PhD.
What sort of background do you expect in a student?
I'll try to explain this in terms of Waterloo courses and degrees, but you can translate into your own institution. I should also emphasize that I'm putting these down not as requirements to start working with me, but as an indication of which directions are useful. I think the Waterloo Pure Math requirements are a good start. The other regularly offered courses I would particularly recommend are Geometry of Manifolds, Representations of Finite Groups, and Introduction to Lie Groups and Lie Algebras; these are common courses to take during an MMath degree. Waterloo also has a Mathematical Physics concentration; for working with me, that's probably not as good a preparation as a Pure Math degree, since it leans quite a bit more toward the analytic/"hard" side of things.
You keep mentioning Perimeter Institute; are you a mathematician or a physicist?
Definitely a mathematician. However, I have a joint appointment between the Pure Math department at UW and PI (which doesn't have departments, as such), which is a physics institute, but an open-minded one. I have offices at both institutions and split my time somewhat equally between the two of them (they are about a 20 minute walk apart). My research at the moment is very much mathematics, and doesn't necessarily require any quantum field theory to understand, but I'm increasingly looking to quantum field theory for inspiration and research directions. My recent talks at Fields (download here) and in the WHCGP (details here) form a good introduction to the kind of physics ideas I'm incorporating.
The very short version of this story is that I became interested on the math side in certain non-commuative algebras and associated symplectic singularities. Physicists kept trying to tell me these objects arise from certain special 3d field theories and eventually I started listening. In the last few years, an important focus for me has been understanding both how to gain insight on the mathematical questions of interest to me from this physical perspective, and also how it can lead us to new questions in mathematics.
My approach to these topics is very "soft": I'm not trying to evaluate path integrals, or solve PDEs. But quantum field theory has deep and beautiful algebraic structure that points to a lot of interesting mathematics. So I try to think of quantum field theory as a guide to interesting problems, which I try to boil down into algebra as quickly as possible.
For a student, I would say no knowledge of quantum field theory is needed to work with me, but some background in it will help, and will allow you to get much more out of the other people at PI. We have a very active seminar which is mostly mathematical, but which it requires some knowledge of quantum field theory to keep up with.
