Introduction

Back in September 2020, I identified my advisors and a project for my doctoral study in the Mathematics Department at the University of Tennessee-Knoxville. I had a very carefully thought out plan for how to identify an advisor(s) that would be a good fit for me given my interests, needs, and career goals. This plan was part of my larger idea of how to become a visible entity in the department as a graduate student relatively new to the Ph.D. program. I told this story here almost a year ago.

Now, I’m reflecting back on that process and realized that I made a very good decision, so I’m going to share some new criteria you might want to consider if you’re looking for a (Ph.D.) advisor of your own. Also, I’ll talk about some of the advantages and disadvantages I’ve realized come from having two co-advisors, rather than a single advisor that oversees everything you’re working on.

Advantages of two co-advisors

Each co-advisor will inevitably have their own research program, even though the process of being co-advised is effectively a three-way collaboration between you and the advisors. What this means is that the two advisors will have different — but likely overlapping — areas of expertise. This gives the student an exciting opportunity to work in an area where those two sub-disciplines merge into something that could very possibly be novel, and at the very least exciting and fresh. With this in mind, the process of generating research questions and directions to pursue should be streamlined. In my case, one of my advisors, Tadele Mengesha, makes his home at UT’s partial differential equations (PDE) group, while the other, Abner Salgado, is part of UT’s computational and applied mathematics (CAM) group. A lot of the theoretical developments we’ve made have been inspired by previous projects that Tadele has been involved in, while Abner has taken the lead in building a road map for how computation and numerics can enter into the picture.

There are also some practical and logistical advantages to the setup of having two advisors. One of these is a faster reply to emails when I have questions between meetings. While a reply from one of them is usually enough to answer my question, sometimes my questions are more open-ended and have multiple valid answers. For instance, I recently asked them about why they favored one proof technique we were considering over another. Abner emailed me back and gave a response that seemed inspired by his background in finite element analysis. While I’m still waiting to see if Tadele has other comments, I imagine he will. As a bonus, this also shows that they are proactive in communicating with me.

As for another advantage, let’s go back to what I said about each advisor having their own research program. One of the consequences of this, is that each advisor will likely be applying for their own grants, and there is a higher probability that I will receive financial support to aid in carrying out the research. Based on discussions I’ve had with Abner and Tadele, in addition to those I’ve had with other graduate students, this can come in the form of summer support, funds for traveling to conferences, or a reduction in teaching load. Of course, the exact details here depend on the funding structure of the program in which you are enrolled.

Disadvantages

Honestly it’s been hard for me to come up with much in terms of disadvantages of having two advisors, especially given that the advantages have been pretty substantial. Just a couple of small logistical things to consider. First, it’s obviously harder to schedule meetings with three people instead of two. I’ve found it hard to run meetings with only one of the two co-advisors, because we try very hard to keep everyone in the loop. If my thesis eventually becomes two separate projects where one advisor is doing most of the guidance on each, this problem would be mitigated.

In principle there is also the scenario where two advisors can disagree on the best course of action at a given moment (what approach to use to solve a problem, for instance). Oddly enough this hasn’t really happened yet, unless the disagreements were anticipated and resolved between my advisors outside of our meetings.

Final takeaways

Regardless of whether you are thinking you’d rather have a single advisor or a pair of advisors (or if this distinction isn’t important to you at all), I want to make a few final remarks on the advisor searching process that I’ve come to observe.

1. Do research to see which potential advisors have track records of securing grants, from the NSF or elsewhere. You may be able to contact past/current students of those advisors and ask if they have been supported financially for conference travel or otherwise.
2. What excites you? Is there a specific area of math you want to dive deeper into? There may be a specific project you want to get involved in, but hopefully you addressed that before enrolling in the graduate program. If you want to cast a wider net like I did, there are hopefully several advisors whose work interests you (otherwise you might be in the wrong department). Talk to each of them, if only informally, and start building those connections, if you haven’t already. Taking classes with these professors also helps with this, if you have the opportunity to do so. Even in graduate school, I’ve base my course selection choices in part based on who is teaching the classes.
3. What are your career goals after finishing the Ph.D. program? If you’re not sure, that’s fine, and as my advisors have suggested, it’s important to keep an open mind. But I strongly recommend being transparent about your thoughts in this regard, because your advisors may have connections that will be valuable to you in your job search, and at the very least, this may help them cater the research direction in a way that would be suitable for you.

As for my specific project, I picked my advisors in part because they shared an important vision with me: a desire to explore both theoretical AND computational aspects of mathematics. I carry this vision in part because I am still unsure if I ultimately want to pursue a career in academia or industry, but also because I’ve come to enjoy mathematics from both a theoretical and a computational standpoint. Even before starting my project, I’ve found great joy in picking apart an analytical structure and understanding the subtleties of mathematical proof. On the other hand, I also enjoy making predictions of how mathematical systems will behave from an algorithmic standpoint and then implementing those algorithms.

Disclaimer: I’ve gone verylight on the technical details of what I’m working on with Tadele and Abner. This is intentional. The point of this blog post is not to describe the mathematics behind what we’re doing, but rather to focus on the operations of how we are moving around the mathematical landscape.