I'm still new to this whole advising business, but I'm usually looking for students who want to work on some cool research. Sometimes I might feel overstretched, and I'll tell you so, but there's no harm in asking!
Try reading my research interests. If it sounds cool (even if you don't understand every word) or even just if you're a visual thinker, then we might be a good fit for each other. You can also browse through my papers to get a sense of what I think about. My interests are pretty broad and I'm happy to supervise a wide range of topics but there is definitely a “flavor”.
Here is what the process generally looks like.
If you're a grad student:
Email me with your interests and background (including but not limited to what courses you've taken at UCSB). Then we will chat about some possible topics for you to learn and think about. You should think of the first quarter as a trial period where we get to know each other. Compatibility is very personal: at the end of the quarter, if it turns out we're not compatible, either of us can decide to stop working together, no hard feelings.
Preparation and background: Generally, for most topics I could supervise, you will need to know some algebraic topology (MATH 221ABC and 232AB at UCSB). The best resource for that is Hatcher's book.
Beyond that, the stuff you need to learn will vary a lot depending on the topic. Many (but by no means all) of my projects use rational homotopy theory. The best resources for that are Griffiths and Morgan, Rational homotopy theory and differential forms, and Félix, Halperin and Thomas, Rational homotopy theory. Some projects might use geometric measure theory, probability theory, or theoretical computer science.
My advising style and expectations: Since I'm still new to this, I'm still figuring out what works and what doesn't. Here are some things that I'm sure about.
- We will set up a weekly meeting time. We might not meet every week, but we both always know it's a time we have open.
- I will generally give you problems to think about early on, and/or encourage you to find your own problems you're excited about. For myself, I find that it's easier to learn a new topic if I'm trying to apply it to something specific.
- I expect you to attend most topology seminars. It's OK if you get lost in most talks at first; try to get something out of them, or ideally three things.
If you're an undergrad:
I can occasionally supervise undergraduate summer research projects or senior thesis projects. If you want to do research with me over the summer, let's get started early: we will do a reading course during winter quarter in order to prepare you and to decide if it's a good fit.
Generally you need to have at least taken the core undergraduate math major courses (algebra and analysis).