Information for prospective and current graduate students

If you are considering working with me, here is some information about my advising style and general suggestions.

What to expect if you work with Lvzhou

Research areas

I would expect my students to learn and work in areas fairly close to my research interests: geometric topology, geometric group theory, topological methods in group theory, and dynamics. These areas have considerable intersections, which is fascinating.

Initial communications

If you are a graduate student seeking a potential advisor, I would urge you to talk to all potential advisors whom you are considering as early as possible. This is helpful for you to know each other, receive helpful suggestions from your potential advisors, plan forward, etc. If you are interested in working with me, I hope you can contact me by the Spring semester of your first year, but as early as possible.

Weekly meetings

The goal of advising is to train you into an independent researcher in math. An important part of the training is done in regular weekly meetings. The length of each meeting is typically about one hour.

Prior to the meeting, you are expected to

have spent enough time reading or working on what I suggested,

tried hard to solve any difficulties that appear (e.g. by thinking hard, searching online, reading other sources, working out simple examples, or finding an alternative way, etc.),

clearly summarized your new findings (or what you learned) and any difficult parts worth discussing.

During the meeting, I would expect you to explain and write on the board with clear statements about what you have found or learned. It is good to bring some very brief notes you have prepared just to remind you what to discuss, but it is usually a bad idea to read and copy from a book or paper. I will comment on any ideas or arguments you present or ask questions, answer your questions, and suggest ideas or references to deal with any difficulties you run into.

After the meeting, I would expect you to write a very brief email to me summarizing what we discussed, which could be helpful for future references. You may also write up some of our discussions (that you find important) in your own more detailed notes.

Different stages

You are expected to do some reading at an early stage to build up necessary background knowledges for your research. During the reading, you may find some problems of interest to work on, or I may suggest some problems that I think are worth working on.

After working out an initial project (hopefully not the last one before you graduate), you are likely to have some follow-up problems to work on or have learned how to find interesting research problems.

When you become more mature, you may work on several projects at the same time and have collaborators. At this stage, our meetings might become less structured and less frequent.

Courses to take

You are always welcomed to discuss with me about what courses to take. Apart from passing the core courses and/or exams, you are expected to be familiar with materials from the following courses: Algebraic Topology (57200), Differential Geometry and Topology (56200), Riemannian Geometry (66100). You are probably required to take some of these courses as requirements for your Advanced Topics Exam; Check the graduate handbook for details.

In addition, you are recommended to take relevant topics courses. Look out for topics courses offered by: Lvzhou Chen, Ben McReynolds, Sam Nariman, Sai-Kee Yeung, Thomas Sinclair, Xingshan (Shawn) Cui, Eric Samperton, Jeremy Miller, Ralph Kauffman, Manuel Rivera, etc.

If there are few relevant topics courses offered in a semester, you may do some reading with me (or others) or assign more credits to independent study (after passing the Advanced Topics Exam).

Seminars to attend

Try to attend the Geometry and Geometric Analysis Seminar and Topology Seminar regularly. Both seminars have talks that I find interesting. At least in the first a few years, it is good to attend every talk, even when you don't know the topic at all: It is a way to learn about research topics in your field or some nearby fields. These talks can be hard to understand in the second half, but you can always get something out of it (e.g. the rough flavor of this topic, what kind of tools this topic might require, how to give a talk that people like, or in the worst scenario: how not to give a terrible talk). Later on, you might have fully developed your own interest, figured out certain topics you definitely don't enjoy, and need a lot of time on your own research, then skipping some of the talks would make sense.

The Operator Algebra Seminar sometimes also has interesting related talks.

By a similar logic, try to attend the colloquium whenever we have one (at least those in pure math). They can be in a less related field but usually more accessible. Job talks can be less accessible at times, but it is good to attend those in geometry and topology.

You are also encouraged to attend and give talks in relevant student seminars, such as the topology student seminar.

What to read

Apart from what I specifically asked you to read, it is great to read other textbooks, notes, or papers on relevant topics based on your interest and urgency (defined by your research goals). See the link below to Katie Mann's website for a wonderful reading list.

Useful links

Excellent notes or books by Danny Calegari,

Great notes on specific topics by Andy Putman,

Suggestions by Katie Mann and a big reading list she wrote,

A mind-blowing blog by Danny Calegari, full of interesting geometry, topology, groups, etc. Unfortunately, it is dormant.

Website for a geometric group theory course taught by Henry Wilton,

Lists of conferences in related areas: Geometric group theory, Low-dimensional Topology,

Other helpful resources (e.g. related to job applications) from the Topology Student Workshop by Dan Margalit.

Other General Suggestions

Here are the big questions puzzling PhD students: How should I do research? How can I become a good researcher (in math)? I believe there is no universal answer, so you need to seek your own answers as you grow. What people can provide are some general suggestions. Below are some suggestions (mostly from others) that I personally find helpful.

(Learn to) be smart

Wisdom can be acquired. One can observe and learn the way of thinking from others, especially those super smart people. Thinking in a smart way is essential for efficiently solving many problems and overcoming various difficulties in research. Many breakthroughs in math are made by finding good new angles. This is very hard to do, but one should always try and also learn from those who are good at it.

Clarity

Experts can often clearly explain a concept, theorem, or a long proof in a few sentences. This kind of clarity is essential to deeply understanding (and memorizing) mathematics. It is an important skill to acquire to be able to extract such key ideas from a long proof for example. To this end, it is helpful to keep asking yourself to summarize the key ideas of proofs that you read about as exercises. The other side of the story is being able to work out the details of the proof from a brief outline, which is another important skill but a different story.

Be broad

A knowledgeable mathematician can recognize the theory related to the (sub)problem at hand and find suitable tools from a field possibly distant from his/her research area. Being knowledgeable plus some good imagination to draw potential connections is the key to finding new angles to attack a hard/old problem. There are many ways for a graduate student to learn new math: taking (topics) courses and seriously digesting the materials, learning from (student or regular) seminars, reading books/papers (suggested by advisors or just of interest).

Work hard enough

This is not to destroy your work-life balance, but it takes a LOT of effort to succeed in mathematics. Around the time when I entered graduate school, I read an article about a speech given by Benson Farb on "How to do mathematics?" [A link to the article along with Chinese translations] The following sentence from the article pushed me to work rather hard during certain periods.

I would say (work) forty hours a week if you want to be an incredibly crappy mathematician. Maybe you can get away with forty hours a week, but eighty hours may be more realistic (to succeed).

My practice suggests that working 80 hours a week (on math) is not so realistic to achieve while maintaining a fairly normal life. It does sound correct that a graduate student should try to work at least 30 hours a week apart from teaching duties (so that would exceed 40 once we include teaching). Well, the actual number does not matter. What really matters is the attitude and to remind yourself that you need to work hard (when you are not working hard enough).

Keep a library of examples

When you learn about a new theorem, it is helpful to apply it to interesting examples to get more concrete and deeper understanding of the theorem. To do so, you will need to know many good examples. Having good understandings of many important examples would build up your intuition. Similarly, when you make a new conjecture/guess as an attempt to solve a problem, it is helpful to check your conjecture on key examples.

Try the simplest nontrivial examples

I find it an easy but useful trick to progress when I get stuck.

Pick the simplest example (among the objects that I try to understand or prove something about) that is nontrivial for me to understand;

Try my best to gain good understandings of this example;

Repeat the two steps above until I get a fairly clear picture (or some good guesses) of the general case.

Part of the process might involve computer experiments, which can assist us in working out complicated examples or suggest undiscovered patterns, etc.

Keep notebooks

I think I have a rather good memory, but I still keep forgetting things. So it is important to keep notes. I would especially recommend writing down:

Summaries of (math) conversations with people;

Things that I tend to forget. For me, these are maths that I don't use or think about frequently (but I usually only write about those fairly close to my research for the sake of time).

Notes can be written in any form, but I personally like electronic ones as they are easier to search (and store).

Current students

Geoffrey Baring