From Rishi Sonthalia (AIM PhD): Once I was admitted, I went onto the math department website and went through each and every professor’s personal bio/website to see what they worked on. Part of this involved not just reading what they said about their research but also looking at their recent publications. (I did this because I feel a lot of bios/websites haven’t been updated, but their recent papers show what they have been interested in.)
I then made a list of every single person I could potentially be interested in working with. Once I had this list, I started emailing professors (Winter first year) from this list asking for a meeting. Sometimes getting a reply took some time. Some people I had to email a second time a few months later to try again. But I did get a meeting with everyone I emailed.
Then once, I met with them I asked them about their interests and whether they would be interested in taking on new students. I did a get a few negative responses, but I then asked these people who they thought I could work with, and then I set up meetings with these people.
I actually found my outside of the math department advisor like this. A math professor I met with gave me 3 names of people to talk to, one of these is my outside advisor.
(Rishi is working with Anna Gilbert)
From Gilyoung Cheong (Math PhD): Coming to Michigan, I was quite determined to work with Wei Ho, whose work seemed to involve many fascinating parametrizations of various arithmetic objects. Despite the matching research taste, after having some conversations, I felt that I was not strong enough to parse through some prerequisites before getting to the frontier of the research that she would like to think about. I ended up not working with Wei, but I still plan to study some of her earlier work in the near future. Moreover, she has given me great career advice at various places like number theory lunch, GLNT (Group, Lie, and Number Theory) seminar dinner, or GROW (Graduate Research Opportunities for Women) conference, so I still think it was a great idea to contact her.
When I was struggling to find an advisor, there was a junior colloquium talk that was organized by Karen Smith, and the talk was given by my current advisor Mike Zieve. Mike talked about Galois coverings of some Riemann surface, and I vaguely remember that he used some result about finite groups to obtain something new. It was quite clear from his talk that Mike did not require heavy prerequisites before he jumps into a research problem. This does not mean that he does not care about big machinery, but his need of studying any machinery often comes after a concrete problem.
What I didn’t realize in the beginning of my graduate school is that not everyone needs to work on big machineries. Even though I like to learn about formal foundations of many mathematical theories (such as commutative algebra and algebraic geometry), I only tend to learn them as I need when I work on research problems, and Mike was the same way. To see if I can be his student, I decided to talk to Mike about some computations I have made about counting roots to some polynomial equations over finite fields. Mike gave me extensive feedback, and I immediately asked him to be my advisor. The computation became a small publication, thanks to Mike’s help.
Oddly enough, I have never “worked with” my advisor Mike Zieve on projects. Although he and I work on different mathematics, I do enjoy listening to him talk about his work (mostly something to do with Galois theory). The topic may not be relevant to my works, but I still observed how a professional mathematician thinks about research problems. Mike also helped me numerous times when I felt like I did not know enough math or my result was not good enough. Whenever Mike critiques any of my writing, it becomes a much better paper. He understands my style of research so well that he often suggests which mathematicians I should talk to, and I have benefitted much from this (e.g., Eric Rains at CalTech).
The bottom line is that people work differently, so I hope you find your style and any advisor you can work with. (Read Karen’s advice above, which seems to be quite onpoint.) Good luck!
From Jenia Rousseva (MLB 2018, currently AIM PhD): I agree with everything written in the “Finding an Advisor” section, but maybe I can elaborate more on the first point for Marjorie Lee Browne scholars. As an MLB student I had the option of starting research during the summer before my first year and I had to find a research advisor by the end of my first year with whom to work with that summer. I would like to point out to future MLB students that they may work with the same advisor or a different person during these summers. Although, as masters students we had less time to find a research advisor, there are still opportunities to explore collaborating with different people and working in different fields. MLB students in the AIM program even have the option of choosing a research advisor outside of the math department.
From Jasmine Powell (Math PhD): Coming into grad school, I had some very vague ideas of areas I was interested in but no particular person or field I was sure about. I took the alpha courses my first year, and the instructor of one, Sarah Koch, was a fantastic teacher and a lot of fun to talk to. I had never even heard of her area of research (complex dynamics), but once I realized that she seemed like she might make a good mentor, I spent some time reading about her area and thought it might be something I’d enjoy as well. The summer after my first year I approached her about doing a reading course, which we did throughout my second year. In that reading course, I discovered that:
 I absolutely loved the math. It was an area that I’d never thought about working in, but it mixed all of my favorite parts of other math that I’d seen (lots of topology, LOTS of pretty pictures!), and
 Sarah was someone I wanted to work with. Her motivational, excited mentoring style was and is one that I think works very well for me, and I realized that she embodied a lot of the qualities of a research mathematician that I wanted to cultivate myself.
Once I figured out those two things, I asked if she’d be my advisor!
From Ursula Trigos (MLB 2018, currently AIM PhD): I used to always say that “I got lucky” with finding my advisors, a sort of rightplace, righttime thing. But looking back, it seems I was making a lot of choices that put me in that right place.
As an incoming first year, I had never seen graduate students before. My undergraduate institution had no higher degree in mathematics. So, upon entering the MLB program here at University of Michigan, during orientation I clearly recall that they sat us down and told us “You have all seen graduate students, you know how they act. This is you now, act appropriately.” I was thoroughly bemused.
To make up for this fact, I attended as many seminars as I could. I reached out to as many professors as I could, those studying meteorology, climate change, biology of large mammals, biology of virus and pathogens, ecology— and I asked as many questions as I could. I reached out to other students in my program, all of whom were very helpful! I even ended up googling topics in math and science that I found interesting (along with “umich”) to see who came up, and I would email that person!
I met with several professors and just asked them about their research and what they were looking for in a potential advisee. I even started working on several projects but nothing really felt right.
Then in an AIM seminar my current advisor, Dr. Annette Ostling from the EEB department, came in to give a speech about topics she wanted to pursue in research, and said she was looking for a mathematician to help. What really struck me about her talk was how everything she was passionate about was so aligned with what fascinated me. Every project she spoke about sounded like something I would want to pursue.
I emailed her after the talk and that was that.
The best advice I can think of would be to open yourself up to possibilities: go to talks you may not normally attend, challenge yourself to think about what really interests you, and pursue it.
From Umang Varma (Math PhD, 2019): I came to UM from a small liberal arts college with a lot less background than many of my peers. I knew no category theory, didn’t know what a quadratic form was, had never heard of group actions, and never seen the definition of a measure. In retrospect, these were the easiest challenges to overcome: I worked hard, talked to my peers, and went to office hours. At the end of the year, I didn’t feel like I was behind my classmates in my alpha classes and passed the quals comfortably.
My biggest struggle in graduate school came in finding an advisor. During my second year, I organized a reading course with a professor who was working on questions that I found interesting, but I often felt uncomfortable asking “dumb” questions. I already knew that other professors in the field wouldn’t be good matches (based on conversations with other grad students and some professors) so I decided to look in a different field.
I picked up papers by a different professor and started reading them. I found the proofs really cool, even though they were hard to understand initially. I started speaking with this professor regularly. About six to seven months later (a couple months into my third year), it became clear that this wouldn’t be a good match. I didn’t feel like I understood why we were trying to answer these questions and it would be a big detour to get to a point where I did. The professor was also upfront about their expectation that students come with questions they want to work on. We agreed that with my background, it would be hard to get there in a reasonable time frame.
I soon felt helpless and unsure of what to do. The two fields that I had tried to get into felt like they wouldn’t be good matches. With one semester remaining in my third year, I didn’t know how to find a different advisor and different field. The one really helpful advice I got from some peers was to try getting an advisor in theoretical CS as it is a field that students are able to jump into pretty quickly (plus, I had some background and interest in CS). Although this process was complicated by poor timing (my first few choices based on their recent papers/online profile were away on parental leave or sabbatical or had recently moved to a different university). However, at the same time, I was asked by a professor in math if I was interested in working on their new project. I believe a couple professors in our department who knew my situation had suggested my name to this professor. I found the topic really interesting and the professor very supportive, and was able to jump in quickly. We agreed within weeks that we wanted to work together and I took my prelims a few months later.
Some takeaways:
1. This process can be long and difficult. Unlike passing quals, there isn’t a simple “work harder” solution to finding an advisor. I’ve seen myself and my peers struggle more with selfdoubt and imposter syndrome during this phase of grad school than any other.
2. There are probably some additional hurdles you face when you come from a small school or come with a less extensive background. You haven’t been exposed to as many different flavors of math and you spent a lot less time taking topics classes because you were focusing on alpha classes/quals. None of these make you any less capable, but it’s reasonable to acknowledge that this might affect your grad school timeline and how you go about finding an advisor.
3. Don’t be afraid to pick topics that are outside the “mainstream” in math. Both I and many of my peers who had some struggle finding a field/advisor have done a lot better once we found advisors who were very supportive and topics that we were really excited by.
(Umang wound up working with Anna Gilbert)
From Trevor Hyde (Math PhD 2019): Finding an adviser is not unlike dating: expect to leave your comfort zone and “make the move”; expect awkward, unpleasant interactions; and expect to court several people until you find The One. I came to grad school hearing that Professor X was *the* person to work with in my area. I signed up for this Professor’s class and it didn’t go well. It was boring and confusing and far beyond my level. I tried to meet with the Professor outside of class but the meetings were awkward and it always felt like the Professor was waiting for me to leave. So then I tried setting up a meeting with the new hot shot Professor Y the following summer. That went similarly: Professor Y would answer questions curtly without elaboration and then sit in silence until I asked another question or left. These were the typical interactions I had with professors my first couple years of grad school–quite different from my experience as an undergraduate at a liberal arts college. This stressed me out but I tried to focus on getting through my quals in this time. In my second year Professor Z returned from several years abroad and I took Z’s course. I enjoyed the lectures and put a lot of energy into the problem sets which the Professor noticed. Professor Z engaged me in conversation after class and I began coming to his office. Finally an interaction which felt (more) natural with a Professor who didn’t act like he couldn’t wait for me to leave! I arranged regular meetings and kept working hard in the class. Eventually I broached the question of whether Professor Z would take me as a student (a conversation that felt very much like a marriage proposal) and he agreed–in fact he had my file on his desk and it seems he had just been waiting for me to ask. The moral here is, I think, to stay hopeful and proactive even when it feels like you’ve exhausted all your possibilities. There are many professors in the department with a wide range of personalities. While it is unfortunately true that many professors seem to actively repel students, there are others who are just waiting for someone to come talk to them. Keep in mind that the department is always in flux with new people showing up each semester! And yes, when you’re single it sometimes feels like you’re doomed to be that way forever, but look around and be reminded that it does seem to work out for everyone eventually.
(Trevor worked with Mike Zieve.)
From Visu Makam (Math PhD 2018):
1. There is a place in mathematics for anyone that wants one. Do it your own way.
2. What you can offer to the subject matters, and not just what the subject can offer you.
3. If you enjoyed a certain course, doesn’t mean you’ll enjoy research in that area. Courses are often highlight reels of the stuff discovered ages ago, so you don’t see the ugly parts. In the subject of research you pick, the ability to get through these ugly parts without losing your mind is nonnegotiable.
4. Think about whether you want an advisor whose strengths are a superset of yours or one whose strengths complement yours. And I don’t just mean strengths in terms of area of expertise. You should also think of style, attitude etc.
5. In the hours you put in for research, go for quality over quantity. Keeping an active lifestyle helps more than you might imagine for this.
6. I would summarize grad school as the place where you learn to fail. Most likely, you have to fail enough, and learn from these failures before you actually solve something dissertation worthy. So, start early and try problems way before you think you are ready for it.
7. If you are having a lot of trouble making a choice, toss a coin. If you want it to desperately land on one side, then you know your answer. If not, then it doesn’t matter. Luck plays such a big role in grad school (and life in general) that you might as well let the coin decide. If you strongly disagree with this approach, find another way to deal with it.
8. Don’t be afraid to reach out to people to help you get through anything. Pretty much everyone who has been through grad school will understand and try to help.
(Visu worked with Harm Derksen).
From Rankeya Datta (Math PhD 2018): I joined UMich from Columbia University, and having studied a fair bit of commutative algebra and algebraic geometry, I knew I wanted to work in one of those areas. After taking 631 with Karen, I was pretty sure I wanted to work with her because at the point of time in my life, I needed someone who would not hesitate to call me out whenever I pretended to know something but really didn’t. I also had the problem that I was (and still am) a very detailedoriented mathematician. In one of my assignments, Karen wrote “it’s hard to find the forest in your trees.” This frankness convinced me that she would teach me how to be a good writer, a skill I desperately wanted to cultivate. After attending her office hours (she usually always has her door open so you can walk right in), I also felt like I could talk to her about things outside mathematics. In summary, I decided to work with an expert in my field of interest who I felt would help me rectify my shortcomings, be friendly, and keep me grounded.
As Trevor points out, asking someone to be your advisor does feel like a marriage proposal, but this marriage is a lot more fluid. You should talk to mathematicians other than your advisor. Of course, it helps when your own advisor introduces you to other senior mathematicians in your field (Karen always does this at conferences), and encourages you to start collaborations with them! So I guess find an advisor who is not overly possessive 🙂
From Siddhant Agrawal (Math PhD 2018): Coming into grad school, I had some background in Analysis after doing a master’s thesis in PDEs and I was also interested in Geometry. I knew I was not so much interested in Algebra but didn’t hate it. I basically made a list of potential advisors I could work within the areas of Analysis (Probability, PDEs etc) and Geometry (more analytic than the algebraic side). I really needed to narrow down my interests so I attended essentially all seminars/colloquium in the areas, took/sat through some advanced courses in the areas and talked to senior grad students. It quickly became clear to me that I did not have enough background in geometry/topology and more importantly I did not like some aspects of the area (e.g. I never really overcame my scepticism of whether some proofs in algebraic topology/differential geometry were really rigorous or not). On the other hand I had the impression (which has been reinforced later on) that areas like Riemannian Geometry are a lot more connected to other areas of math than say the area of PDEs. Nevertheless I decided to not work in Geometry as I did not like to think about the mundane/day to day things in that topic and only really liked the big/major/cool things in the area.
After some time I had narrowed my interests to Probability and PDEs (the fact that there were enough potential advisors in these areas was also part of the criterion). I started talking to senior grad students and tried to figure how working with the advisors would be like. After making a new list of potential advisors I started to seriously consider different aspects such as
 Does the faculty give a problem to their students
 How frequently they met and how much time does the faculty give to the student
 How supportive the faculty is both academically and financially.
 How active they are and the kind of problems they are working on right now (by looking at recent papers and talking to older grad students)
 How their previous grad students did after their PhD
I had then narrowed down the list to 3 advisors and then I just chose my advisor because I liked her personality and her way of doing math (from the observations I made while attending the PDE seminar). I started doing some reading over the summer with my advisor and really liked how it went so I made it official.
(Siddhant worked with Sijue Wu)
From Rafe Kinsey (Math 2014): It’s important to get advice from lots of people (grad students both your age and older; PhD alumni–you can find them with google, math genealogy, LinkedIn; professors; postdocs). Some people’s advice will be better than others, but it’s hard to know which ones are best, so triangulating from a lot of sources and weighing more heavily the ones that seem wiser is a good approach. Also, people’s advice will be affected by their experiences, and its relevance might vary based on your needs. (For example, someone who’s very thickskinned might have different advice about choosing an advisor than someone who’s more sensitive, and where you fit in that spectrum affects whose advice is most relevant to you.) Getting advice from a lot of people also has other benefits: it helps you develop relationships/a network, which is crucially important in life, whether you stay in academia or go onto another career.
Selfknowledge is important and difficult. By the time you finish your PhD, you’ll have a good sense of what type of math you like, what working style you like (frequent meetings vs not, handholding vs independence, friendly/convivial vs focused on business, collaborative vs “tell me what you did”), etc. The challenge is to have a good prediction of what that will be 34 years earlier, when you’re making an advisor choice. In your first 12 years of the Ph.D. (from day one, really, even before you start the official advisor search), focus on that selfknowledge. For example, when I was choosing grad schools, I sensed I might like either analysis or topology. Within my first year, it quickly became clear that I was destined to be an analyst–I love estimates, and I discovered that I actually didn’t particularly like topology. This let me focus my advisor search on faculty in analysis and PDE.
Talk to many professors. I probably talked to 6 or so professors, across analysis and PDE. The analogy with dating is apt, except there is an asymmetry: you might be nervous and inexperienced at this, but the professors will generally be understanding and supportive. They know that you, a young grad student asking to talk to them, are interested in them as a potential advisor. It is fine to “date” multiple potential advisors at the same time for a short period. For example, ask 24 professors (from a larger list of 48) for suggestions of reading, and meet with them over a few weeks, and then start narrowing down the list. (If you’re around, spring/summer after first or second year are good times for this.)
One thing I’d encourage is that you try to maintain some of these relationships even after you choose an advisor. These professors can offer suggestions, serve on your committee, write letters for you, etc. Remember, networks and connections are valuable. Also, as you move beyond your Ph.D. project to a postdoc, your research interests will broaden, so it’s good to have some of that breadth of connections and exposure to related areas of research while you’re in grad school.
One consideration that might be material if you’re focused on staying in academia is your advisor’s success placing students. (This didn’t apply to me since I left academia, but I saw it for my peers who applied for academic jobs.) To first order, focus on an advisor whose area of research you’ll love and who you’ll be productive with–that’s most likely to maximize success, since you’ll be most likely to produce good research. That said, some advisors are better than others at getting their students to advance in academia. This could be because their area is hot (which might change in 5 years) or at least not cold (some beautiful areas of math are no longer as lively, and you might not want to work there), because they have a strong reputation, or because they’re just good at getting their students positions (e.g., because they have good networks). Still, this should be a lowerorder consideration; there are plenty of examples of advisors without a long “track record” placing Ph.D. students who have subsequently had students who’ve done well, and you’re much bettersuited with an advisor you click with than choosing someone just because their students do well. In other words, don’t choose an advisor because they place students well, but if you’re split between two advisors and one is really good at this and one isn’t, it might be a deciding factor, and if an advisor works in a dead area or is known to be particularly bad at placing students, that might be a red flag. (Of course, to the extent that you plan to leave academia or pursue a teachingfocused job, this becomes much less relevant.)
Be proactive in the advising relationship. After a year or so of working with my advisor, I realized that I was more productive when I had the structure of a fixed weekly meeting. I suggested this to my advisor, who was happy to do it, and I became more productive as a result. (Of course some advisors might have a specific preference and not be as flexible. This is one of those selfknowledge things. Do you want/need structure, do you want/need freedom, etc? Advisors differ. Similarly, some advisors are willing to spend a lot of time with you–mine was incredibly generous–whereas others have more limited availability. What will work for you?)
Don’t be afraid to switch advisors. I had a great relationship with my advisor, but I had peers in grad school who had difficult relationships with their advisors. Some eventually switched advisors (often rather late); some didn’t but perhaps should have. It happens, it’s okay, people move on. If you sense things aren’t working out, perhaps speak in confidence to one of the associate chairs (e.g., Karen Smith) for their advice. In this situation, it might make sense to be diplomatic and discreet as you explore the possibility, so at not to jeopardize relationships. Of course, math research is hard, so it is important to be gritty/resilient. Try to figure out whether it’s math research and/or you that’s causing the frustration (which is okay, don’t beat yourself up about it!), or whether it really is an issue with your advisor (or that research area).
(Rafe worked with Sijue Wu)
From Luis NúñezBetancourt (Math PhD 2013): When I arrived to Michigan, I had already decided that I wanted to work in Algebraic Geometry (AG). I also had possible advisors in mind. However, during my first semester at Michigan, I took a wonderful class in Commutative Algebra (CA) with Mel that changed this idea. The way Mel explained proofs and motivated theorems made a big impression on me. I also enjoyed his jokes. Our interactions during office hours were really nice. I remember that during the last week of class, Mel mentioned that he was very busy as he has been recently appointed as chair. For that reason, he would be taking fewer students than usual. At the time, I was not sure about switching from AG to CA, but I did not want to miss the opportunity to work with Mel. So, I went to the last office hours of the semester and talked to him. I mentioned that I was not sure about working in CA yet, but that I did not want to lose my chance. Mel was very understandable and gave me time to think about it. He also met with me once in a while to discuss math. I also talked to several of his students. After a few weeks, I ended up being informally his student (I needed to pass my QRs to formalize this). The experience as his student was amazing. I feel lucky to have had that class with him during my first semester (otherwise, I may have done something else). From my experience, the main advice I can give to students is to not be afraid to talk to potential advisors and to get to know them a little before choosing one.
From Sarah MayesTang (Math PhD, 2013): When thinking about graduate schools, I had a couple of excellent advisors who recommended that I think about potential advisors before applying to graduate school. They said that I should choose the person over the specific subject. So, I did a lot of research online about advisors before choosing a grad school, and then really focused on understanding professors rather than professors’ research when I arrived. In my mind, I had two professors who I thought I’d work well with.
I was very fortunate to be assigned Karen as a firstyear doctoral committee advisor when I arrived. Our first meeting was memorable: she dispensed so much useful advice and showed a welcome and open attitude towards math. From that point on, I knew I wanted to work with her. I continued to do my due diligence, however, getting to know other professors in the department and talking to upperyear students. This research mattered a lot less in the end than the nonmath components of the advisor relationship. Karen supported her students, cared deeply about advising, and was an excellent teacher.
While I think that students should consider the broad area that they want to work in (e.g. algebra vs. analysis vs. topology), after that I encourage them to think mostly about how they will work with their advisor and how they will be supported. You need an advisor that you will be able to learn from and who will support you; a good advisor can supervise projects that lie slightly outside of their research interests and inspire you to become passionate about their research interests!
From Sara Lapan (Math PhD 2013): I was an undergraduate at University of Chicago and came to UM thinking about going into algebraic topology, but after just a short time at graduate school I realized that I really did not know what area I wanted to study. After about two years of graduate school, I spent some time reflecting on what topics I especially enjoyed learning about, but I still wasn’t sure what area I wanted to go into. It seemed like such a commitment! With my experiences now, I would like to go back to myself and say this choice on a research area is really not as important as I thought. Choosing someone you can easily talk with is much more important; anyway, your topic choice doesn’t have to be “forever” and in fact it is even valuable to stray from the topic to do research in other areas, too. At any rate, I eventually decided that I was interested in complex analysis in several variables, so I talked with a number of graduate students who already had advisors in the area to get advice on who would be good to work with. This advice from graduate students was extremely useful. From these discussions, I found that some of their advisors would not be a good fit for me and that some of their advisors sounded like they would be. The summer after my second year, I did a reading course with the two professors I was then considering as advisors. From these reading courses, I decided that I could work well with both professors, but I liked one of the topics (complex dynamics) better than the other and so I chose to work with Mattias Jonsson.
From Andrey “Kurt” Mishchenko (Math PhD 2012): My path through the PhD was somewhat atypical so I won’t draw from it directly here. However, I feel like I’ve learned a lot both during my PhD and afterward in industry and beyond, so I’ll write the #1 thing I feel I would focus on if I were to go through the process again.
Good work means collaboration: you can benefit directly from what others have done (using their results, avoiding dead ends), you can get a sense of what is important to other people by reading their work and talking to them about it, and having a rich and broad mathematical vocabulary has a compounding effect where you simply waste less time bored and spaced out in lectures, seminars, and even informal conversations because you are missing some piece of knowledge that the speaker assumes you have. To write a thesis you will have to understand some narrow field deeply, but I think it’s worthwhile spending some portion of your time continuously broadening your horizons, in math and also elsewhere in life.
I think early graduate students often imagine that the progression is something like (1) learn the basics (alphas) and maybe take some 600level classes, (2) pick (somehow) a field that looks “interesting” to you, (3) pick (somehow) an advisor, (4) work on a problem under the leadership of that advisor. For a lot of people that’s indeed how they progress through the degree.
Personally, if I were to do it all over, I would try to (not in any particular order) (1) check out a lot of seminars, and try to understand what they will be about by reading some of the papers beforehand, to maximize my ability to actually follow and get something out of the seminar, or perhaps to not attend because I understand that I’m not going to get anything out of it, relatedly (2) try to understand what the problems are that researchers are actually working on, and try to actually make a dent where I can without waiting for the magic moment when I am a mathematician empowered to do so, which leads to (3) collaborate with people and get feedback on my ideas, have joint projects, have lots of informal conversations. Some of the people from (3) will be professors, and one of them (in my idealized world) ends up naturally becoming my “advisor”.
In short, I would jump right in and try to be a mathematician, and try not to worry too much about the technicalities of finding an advisor per se.
Maybe my advice should be prefixed with, “pass the quals, then…”
(Andrey worked with Jeff Lagarias)
From Emily Witt (Math PhD 2011): Before starting grad school, I had a specific field of interest in mind. However, as an undergrad, I had not delved deeply enough to have an idea of what research in this area might be like. During my first year at Michigan, along with the qualifying exam prep courses, I took a topics course in my “chosen” field. Although it was a good course, I very quickly realized that I was not particularly excited to pursue the area much further.
Since at that point I had passed the qualifying exams, I reflected on my interests during the summer before my second year. I decided that I was most drawn to algebraicflavored material, so I made appointments with several faculty members in algebra and algebraic geometry to talk about their research. Each gave me a brief description on some of the topics in their current research, and at least one leant me a book to look over. Overall, the most effective meetings seemed to be ones with faculty whom I had taken a course with.
The following semester, I asked to take a reading course with a specific faculty member. Not only was the topic especially interesting to me (that specific topic ended up being the focus of my thesis), more importantly, I felt very comfortable in my working relationship with the professor. After the course ended, I asked him to be my advisor, and I have been very happy with my decision every since. My advisor is extremely generous with his time and expertise, helped direct me to focus on interesting and relevant problems, and suggested a diverse array of techniques for me to learn to attach these problems. He is a good friend of mine to this day!
Some recommendations I would give, based on my own experience: Get to know as many faculty members as possible, as early as possible. Don’t be afraid to change your research area from what you originally thought you would study! Talk to faculty about their recent research projects. Consider a reading course with a potential advisor before a studentadvisor commitment is made. Tune into the ways you and a potential advisor are, and are not, compatible: For instance, notice whether there is good communication, on both sides, on math, and on relevant nonmath topics.
(Emily worked with Mel Hochster.)
From Daniel Hernández (Math PhD 2011): I came to Michigan from a large private university, and I moved to Ann Arbor two months before the start of my first year, with the goal of preparing for the qualifying exams. It was then that I first got to know my eventual advisor, though our interactions then mostly involved them providing general advice concerning grad school.
During my first semester, I ended up taking a course with my future advisor. I immediately liked their mathematical style, and I felt that they were someone I could learn a lot from, and also work well with. I continued to take courses in the same general area, with different faculty members. Though it probably should have been obvious, it took me until the end of my second year to approach them about working together.
Before choosing an advisor, I recommend thinking hard about your values. For instance, what is it about the people you like working with (on math, and otherwise) that makes you like working with them? What is it about the people you don’t like working with? What do you think you might need to succeed in graduate school? In fact, how will you measure success in this context? How much do you value personal fit? Mathematical fit? What types of problems do you enjoy working on? Concrete ones? Ones that require more machinery? What types of problems don’t interest you as much? What might you want to do after you graduate? What skills should you develop to make this happen?
When figuring all this out, you should be completely honest with yourself, and realize that your answers may differ (sometimes, drastically) from those of your classmates and friends.
Once you’ve thought about this, you will still need to identify someone that you might be compatible with. One way to do this is to take courses with different people, and also reading courses with a few that you might be more serious about. You will probably find that there is more than one good fit. You should also keep an open mind, and realize that your ideal candidate may not agree to work with you. To measure personal fit, I suggest trying to get to know faculty outside of class. A good place to do so is at tea time, or at departmental colloquia and seminars. When I was a student, there were also semiregular wine and cheese social events, and I met a lot of interesting faculty and postdocs at these events.
(Daniel worked with Karen Smith)
From Kelli Talaska (Math PhD 2010): When I started graduate school, I knew that I loved combinatorics, but I was open to exploring other areas of math. I didn’t know anything about Sergey Fomin, who would later become my advisor, but I saw him at a conference the summer before my first semester and found him extremely intimidating. However, after taking several courses with him, it was clear that this was the kind of math I wanted to do and that he was someone whose mathematical values lined up with mine — e.g. finding beauty and patterns and connections in math, communicating clearly, and finding problems that were doable and important. I remember being quite nervous about asking him to be my advisor, but everything worked out fine.
Personally, I think it’s important to find an advisor you will look forward to meeting with and who will be enthusiastic about mentoring you. It’s also important to note that your advisor shouldn’t be your only mentor — it’s really healthy and helpful to develop and maintain relationships with other faculty, postdocs, and students.
From Felipe A. Ramirez (Math PhD 2010): My “How I found my advisor” story is proof that you can find a great advisor (as I did), even if you are not so methodical and clearheaded in your search for one (as I wasn’t).
I came to Michigan thinking, or at least saying, that I wanted to study Differential Geometry. It had been my favorite upperlevel course in undergrad. But it only took me a short time in graduate school to realize that I was totally underqualified to really know whether I preferred differential geometry over other subjects, or if it just happened to be the most specialized course I’d ever taken. (Now that I’m a bit more qualified, I can report that it was the latter, although I do still enjoy differential geometry.) Indeed, my first year of graduate school taught me that I was underqualified in a host of different ways that I could not possibly have imagined a year earlier. Everything was more difficult than it had been in undergrad, and I noticed that most of my peers seemed to be much better prepared for it all than I was. As a result, I became mathematically reticent; I tended to speak only to people with whom I felt I could hold nonmathematical conversations. I worried that if I talked to anyone too much about math, then they would tell me—indirectly, or maybe through their tone—that I didn’t belong there. Meanwhile, I hoped that I could somehow “catch up” and become “good enough” to belong there. I resolved to persist quietly toward a PhD in Differential Geometry (whatever that meant), and let reality—not people—tell me one way or the other whether I should be there.
Luckily, the professors at Michigan were extremely welcoming, and I felt quite at ease with them as long as I didn’t have to pipe up about math. Eventually the time came for me to overcome my insecurity and approach a potential advisor. By now, I’d taken a number of geometry/topology courses, and I’d read about the research of Michigan’s geometry/topology faculty. Of course, I hadn’t actually talked to any of them about their research (because, well, you know). So I narrowed things down based on how I’d felt in their courses, and whatever I could glean from reading their work. Plus, I still had it in mind to try to become a differential geometer. At one point, I was deciding between approaching Ralf Spatzier, whose survey on rigidity theory I had read and enjoyed, if not absorbed; and approaching Lizhen Ji, whose books on compactifications of symmetric spaces I found intriguing. Not to mention, I loved the courses I had taken with both of them. Still, I didn’t feel qualified to make a truly informed decision on mathematical grounds, and anyway I had long since accepted the philosophy that there are no wrong choices—that I should just make one. So I was basically waiting for something to sway me. The nudge came from Karen Smith, who one day at Tea mentioned to me that Ralf had a good opinion of me. I was surprised, flattered, moved. So I asked Ralf if I could do a reading course in differential geometry with him over the summer, and he kindly agreed.
I got a lot out of the reading course (to this day, I feel like I have some small expertise in differential geometry because of it), and I continued meeting with Ralf into the next semester. I wasn’t sure (and am still not sure) whether I was supposed to formally ask Ralf to be my PhD advisor, and I don’t remember ever doing it. But eventually, he was my advisor, and I was his student, preparing for my prelim exam on Differential Geometry and Ergodic Theory & Dynamical Systems. Ralf had another student (Dave Constantine, now my colleague at Wesleyan) who had already started working on a problem in differential geometry, and I think it is for this reason that Ralf steered me more toward dynamical systems. In the end, my dissertation was about a part of dynamics that is closer to Lie groups and representation theory (and even number theory) than it is to differential geometry. Since then, I’ve migrated further still away from differential geometry, toward the area of interaction between dynamics and number theory.
Looking back, I know it was unwise for me to worry so much about revealing my inexpertise to people. And anyway, maybe I wasn’t as unprepared compared to everyone else as I thought I was. But I also know that these kinds of insecurities are common in graduate students, and difficult to overcome, so I can’t fault myself too much! (At least I didn’t mistake my inexpertise for inability.) One thing I certainly do not regret is having made partially uninformed and naive decisions, and seen them through. I always felt that I was somehow continually lucking upon all the right choices, and eventually I came to believe that there simply weren’t any wrong ones. I still feel that way. On one hand, I’m convinced that if I’d made different decisions, things would have turned out okay. On the other hand, I wouldn’t trade my graduate school experience—least of all my advisor!—for anyone else’s.
From Brian Wyman (Math PhD 2010): I’ll make a few assumptions:
 Your professional network is incredibly important in life/work/etc. You may not appreciate this now, but access to people is amazingly valuable. Many of you – like me – are introverts, even if – like me – you’re a pretty social introvert. Building new relationships may not be natural to you. Your advisor can help to bridge that gap, sometimes even outside of academia.
 You’ve got an idea as to whether your goal is an industry or an academic position – or at least which avenues you want to leave open. Not everyone will want an academic (or industry) position. You need to be clear with yourself about your next step. Or if you’re not sure, you want to consider strongly which doors you want to leave open or closed.
 You have a rough idea of what kind of math you want to do. If your passion is algebraic geometry and you want an academic career, there may be some combinatorics crossovers, but you’re probably not going to work on signals processing or PDEs.
Someone once said to me that choosing a career is a giant optimization problem: who you work with, how flexibly you choose your geography, salary ranges, what kind of work you do, worklife balance, travel requirements (opportunities?), how meaningful (in a humanistic sense) your work is, and so on. Everyone’s objective function is different, so choosing a career can have many different optima depending on how you weight the parameters.
Choosing an advisor is a similar type of optimization. Here are some of the parameters:

 How well do I get along with this potential advisor?
 On a personal level?
 On a professional level?
 How well do our mathematical interests align?
 How quickly can I graduate?
 How much publication am I likely to do?
 In which journals?
 Can I turn this work into a compelling job talk?
 Even in industry?
 Are there natural extensions of this work that can form the basis for several papers while I’m junior in my career and need to progress toward tenure?
 How well do I get along with this potential advisor?
Some of these answers you’ll know right away. Some you’ll need to learn from other grad students, especially older ones. Some you’ll need to talk to the potential advisor about. Some you’ll want to talk through with a trusted nonadvisor faculty member.
Ok, so all of this being said, here’s some general advice:
Understand faculty research areas. Not “algebraic geometry” – what do they really do? There is a layman’s (or rather, a lay mathematician’s) explanation for every. single. faculty member’s work. Find it out. From students, or talk to the faculty member. Which brings me to:
Own your own grad school career, including finding an advisor. Advisors don’t just “happen” to you. You need to get informed, set your goals, and work to achieve them. Advisors – official or unofficial – and friends are there to provide guidance. You should listen to them and understand their perspectives. But at the end of the day, your life choices are yours, and you are the one that lives with the outcomes postgraduation. So, when it comes to choosing an advisor, a research problem, a courseload, a career, etc., you need to own it, even if you (appropriately) weigh others’ advice very heavily along the way. This means proactively seeking information in order to make an informed decision.
Graduate school is super hard. Find an advisor that makes it better, not worse. Not the math, although also the math. Your health, both physical and mental, should be your number one priority. Much has been written about imposter syndrome and other difficulties of graduate school, especially in a place like Michigan where all your peers are incredibly smart and successful. I won’t drone on about this (though, spoiler alert: you are super smart and successful also. Your peers and the faculty recognize this, and you should too.), other than to say that your advisor should be a net benefit to your life, not detract from it. You don’t need any more obstacles.
I didn’t go the academic route, so I will leave some of the more academiaspecific advice to those who did. But for those going the industry route, my advice is:
 Graduate. Find an advisor that will ensure that you do so quickly.
 Learn to give talks. Good ones. Find an advisor that will help you develop communication and presentation skills.
 As a subset of this, do some math that is easily described to others. They don’t have to understand the math, just the big picture. Back to the “communication” point, if you can describe the impact or result of a math PhD thesis to a nontechnical audience, you’re well ahead of most of the technical people applying for jobs.
 Get a job. Seems like a good idea, right? Does your advisor have students who have nonacademic jobs? Nonacademic friends hiring? Probably these questions are good to know the answer to.
My last piece of advice:
 Seriously, don’t sweat it too much. I mean, it’s a really important choice. But it’s also one of many choices you’ll make throughout your life. It’s just an advisor. You can even break up with your advisor if you’re not on the same page. And get a new one. At the start of your fifth year. And still graduate in five years. Trust me. I did this. But, I really don’t recommend it.
(Brian worked with Mike Zieve.)