Introduction

Scroll through the ArXiV on a typical Wednesday night and you’ll see all sorts of papers that are each a culmination of many months of effort, lots of dead ends, and many revisions. Let alone the future steps to push the paper drafts to publishable quality, and many months of waiting for referees to look over the papers.

September 8 marked the one-year anniversary of my commitment to a specific thesis project and a wonderful team of two co-advisors (see my earlier article for the story of how I found this team).

Since I started working with them, the workload has grown exponentially, to the point where the amount of time I spend working on research is the greatest amount possible so I don’t either burn out or ignore my coursework and teaching responsibilities.

Perhaps this is more than I need to do to stay on a pace that my advisors are satisfied with, but I am trying to make the highest-quality thesis possible, leaving no stone unturned. One of my advisors said, “being a Ph.D. student is really a full-time job.” It has taken me a year to fully realize and embrace that sentiment.

In this article I’m going to talk about why my research really is all-consuming, breaking my project into the different types of tasks I spend time on.

What I spend my research time doing

To convince you of how much time and effort research can take, I want to catalogue the specific tasks and activities that my research is divided into. That will make it even more clear that I have a massive undertaking in front of me.

Reading: one of the tasks that takes up the greatest number of hours, reading, reading, reading. Sometimes my advisors ask me to read specific things, such as papers or sections of textbooks.

The purpose might be to fill a hole in my understanding (for instance, I never took calculus of variations because the course didn't have enough demand at the University of Tennessee to run), but more frequently, I’m reading papers that contain proof techniques we may want to use for our project. This often creates a chain where I have to backtrack through citations to reach much older papers.

If one paper references another, I sometimes have to go to the other paper to get a full understanding of the subject matter at hand, and this quickly propagates to me having to refill the paper trays in the printers on campus.

Not all the things I read are “assigned,” though. If I become aware on my own that I’m missing some knowledge I need, I’ll often go and try to find sources myself that can help with this.

Usually this is a combination of performing Google Scholar searches, perusing faculty websites, and digging through the stack of math textbooks I keep in my apartment.

Seminar talks occasionally provide inspiration as well; sometimes I keep reading even if I come to the conclusion that what I’m reading is not of immediate use for my project. As the professor for my neural networks course recently told me, “you’re putting tools in your toolbox; you never know when you might need them.”

Writing: the ultimate tangible goal of [any] Ph.D. is an extensive thesis, where you really demonstrate your expertise on a very narrow topic. I’ve already begun writing, and I don’t even know what the finished product is supposed to look like.

Along the way, my advisors and I hope to get several papers out of the deal whose content overlap with that of the thesis. Thus I’m getting myself closer to these milestones and my coveted degree, each and every day.

In addition to that, constantly writing forces me to improve my understanding of the ideas I’m working with, and writing things carefully has helped me catch mistakes, ranging from minor typographical errors to major logical jumps.

Occasionally I have to completely scrap a proof and start over, or I realize a result that I initially thought was true is actually false. This is definitely frustrating and may make it seem like a whole morning was wasted, but at least I catch it before I go on to prove ten other things that depend on the rickety steps.

Finally, this category involves editing. After a meeting, I often need to revise proofs I have written up, either because they contain mathematical mistakes, or because my advisors think the arguments need to be explained more clearly. This editing is very much a continuous process. Sometimes I go back and edit proofs two months after I thought they were “done.”

Expository notes: one other thing I am trying for the first time is writing some expository notes on a topic that I’m trying to learn more about. That topic is called gamma-convergence, and it is a special type of convergence of functions that, when set up correctly, will preserve minimizers in the limit. So why take the extra time to do this? Well, I was already doing a lot of reading on this given topic, and figured it was worth while to put my own understanding of the topic in writing;

I also worked out some examples that came up in the reading but weren’t fully explained, in order to be more convinced that all the details checked out. As an added bonus, when the notes are done, I can share them for the benefit of others.

Discussions: I’m lucky enough to share an office with other graduate students who work in similar areas to my own. As a result we often bounce questions off of each other related to our own research problems (which makes the office assignments very important).

Sometimes these discussions are premeditated, but more often than not they’re spontaneous. Since I moved into this office in August, at least one idea I got from those discussions has helped me prove a result in my own research.

Dead-ends: research isn’t easy; in reality it’s a perplexing labyrinth that might not even have an exit on the other side! I have to give myself the luxury of dealing with dead-ends at a relaxed pace.

Sometimes it’s better to pause my work and go for a walk or move to a different task entirely because banging my head on the table will just be a waste of time. In extreme cases I’m genuinely stuck and have to wait until I can meet with my advisors to discuss the matter, but often emailing them or looking in a book will be enough to get myself un-stuck.

And SOMETIMES, I just need more coffee. Luckily I don’t drink so much that my body doesn’t feel its boost. I feel bad for those who have crossed this threshold.

The “Friday night” adventures: a few months ago, I read an article (which sadly, I could relocate) that encouraged researchers to block off a chunk of time on either Friday nights or Saturday mornings to think about the most ambitious of the questions that might be produced from research. In my case, that often boils down to conjectures I’ve come up with on my own, as opposed to those suggested by my advisors.

While I’ve found it difficult to make this the same EXACT time each week, I’ve been managing about two hours each Friday evening. Throughout the week I keep a list of problems or questions that might be worth using this time for, then just decide how to use the time in the moment.

On the occasional event where a question I pose is genuinely interesting, I ask my advisors about it. Unfortunately we generate questions to study faster than we can answer them! But I’ve realized this is one of the good problems researchers can have, and then we have to judiciously decide where to focus our attention.

NOTE: Terry Tao says a few words of his own about this matter on his blog, taking the perspective of considering what proportion of time to spend on the most ambitious questions.

Oral exam preparation: In November, I will be taking my oral examination. For this exam, my co-advisors will be joined by another faculty member in the mathematics department as I present the work I have accomplished to date.

The goal of this presentation is two fold: to summarize the results I have proven up to this point (which are mostly theoretical), and to gauge how well I can anticipate and respond to questions about the themes of the work and why it is important. With that in mind, my preparation comes in three main steps: building the slides, brainstorming questions I may be asked and preparing answers, and rehearsing the presentation. I have been working on the first two steps in parallel, and will move to the third one in about a month.

Compared to the written preliminary examinations I took in 2020, the stress level is lower, and the pass rate much higher, but this examination serves as another opportunity to assess my understanding of the research material I’ve been working with, and a thought-provoking question might lead to some new inspirations for the project, if I’m lucky.

The weekly meeting: Finally, there’s the Wednesday afternoons/early evenings where I sit down with my two co-advisors and we talk. After addressing any administrative concerns (e.g. building the committee for my oral exam), we jump into the mathematics, reminding ourselves of the longer-term trajectory of the project as much as possible to motivate the steps being taken at the present moment.

If I am stuck on what I’ve been working on or have questions, I bring those to the meeting; on the other hand, if I’ve worked some proofs to completion and gotten results, I bring those proofs and let my advisors critique them.

In either case this typically involves extensive writing on the whiteboards in the office and careful inspection of an Overleaf document I’ve been writing up that will eventually become my thesis.

We sometimes get carried away at this point, trying to not only flush things out in as much detail as possible, but also discussing any side tangents that might come up, often as a result of initial mistakes or misconceptions that are later corrected.

At the end of each meeting, we make a tentative plan for the next week, with regards to what I should try to accomplish, whether it be reading specific sections of books/papers, working out more proofs, or a combination of both. Later on I will likely have computer code to write as well.

Ultimately my research work schedule largely revolves around the weekly meeting. For instance, if I have short questions I can email them to my advisors throughout the week, but some questions I’ve realized are best saved for the regular meeting time. Moreover, I have to keep track of the clock; while it might be fun to completely immerse myself in research, I don’t want to miss a meeting as a result!

The Consequences

So at this point you’re convinced that research is a very mentally consuming activity, which in my opinion, is much of the point of a Ph.D.

Especially with people saying that your attention gets divided into more and more pieces as you progress through an academic career (for those who wish to stay in academia), having a large amount of bandwidth to devote to one project feels like a blessing that I may never have again. Even my advisors have verified this claim, one of them just being granted tenure a month ago, and the other not far behind.

In this spirit, they didn’t particularly favor the idea of taking an internship next summer, even if I’d be working with a group that did work similar to what I am working on at the university.

Their rationale was that I’m still new enough to my area of research that stopping or otherwise significantly slowing down my progress with them to put my energy into a different project would cost me more than the three summer months, because I would lose all of the momentum I have been gaining recently.

Moreover, this could delay my graduation without good planning. With that in mind, my tentative plan for next summer is to work full-time on my project with them, without taking time for a summer teaching position or other job. This is actually what I did last summer, and I covered a lot of ground, which was important for getting to the point to be ready for my oral exam.

Side projects?

This has been a subject of continual debate between my advisors and I. Much more so than when I was an undergraduate, opportunities come in my path and I need to decide whether they are worth the time investment. Often I go to my advisors for input on this, and sometimes they strongly encourage me to take on an additional project, and sometimes they really frown upon it. So what gives?

I recently realized that it really, really depends on the nature of the project. They know that my intent is to either pursue a postdoctoral position at a university, or a research scientist position at a lab. In either case, research would be where the bulk of my bandwidth would go.

So they only really encourage me to pursue additional projects that would support my research prospects, even if they aren’t directly related to my thesis research.

For instance, I have been mentoring an undergraduate (who is now himself starting graduate school) through two separate projects. The first project was in enumerative combinatorics and lead to three conference talks and a paper (with the two of us as joint authors on everything); the second project is related to some problems in geometry and physics and is still actively in progress.

On the other hand, if the project is related more to teaching or service, my advisors generally have discouraged me to take extra time to undertake the project on. Ultimately they leave it to be my decision, but they have both advised other Ph.D. students before I came along and have a sizable amount of experience to go off of (including their own times as Ph.D. students).

So, more often than not I listen to their advice and try not to overthink it. This is easier said than done for me because I have a tendency to overthink a lot of things, even the more mundane things like what to eat for breakfast.

How to handle the time commitment?

At last, we come to how exactly I combat the large amount of time required to fully dive into research. The first and most important aspect of my strategy is to embrace the work I am doing: to remind myself how much I enjoy it, to remind myself that I am trying to push the boundary of human knowledge, and finally to remind myself that this doctoral degree is the training ground for a full-fledged research career.

It sounds simple, but mentality really matters, as evidenced by the proportion of students who start towards doctoral degrees and never make it to the end.

Now, as far as a concrete strategy is concerned, I need to remember that my research is the most mentally intensive thing I currently have on my plate, so I want to delegate that to the longest blocks of time I have available.

I’ve found, through some trial and error, that I’ll be more productive in research with one three-hour block than three one-hour blocks (and maybe even with four one-hour blocks). That’s because it takes me up to an hour to get into a “groove” and ramp up to maximum efficiency (see my article from last semester for more on this).

As for the points in my week where I have less time to get something done, my running research task list comes in handy. If there is a task that I believe I can do pretty quickly, I very well might try and get it done in that hour. Otherwise, I have other things to work on…

The other part of boosting my research productivity is to contain the time spent on other tasks, particularly teaching and grading. My actual contact hours with the classes I teach are fixed, but I can control when I do my grading, for instance. This semester, I am trying to do my grading late at night, right before bed, for two reasons.

One, it lowers the chance that I get fired up with my research right before I plan to sleep; and two, I’m encouraging myself not to dawdle with the grading since I’m probably getting tired at that point.

Finally, there’s homework. I’m taking three classes this semester, and luckily only one of them has a substantial amount of work (theoretical problem sets and programming assignments in MATLAB). This just gets filtered into whatever time is left in my week, which usually means weekday evenings and the weekends.

All of this work may seem like a lot, and by no means am I trying to pretend that it’s not. Not only do I think it’s manageable when the right strategy is taken, but I get bored easily if I don’t have work to do. No wonder my friends call me a glutton for punishment.