You need lots of mentors, with many different roles. Mentors can teach you new mathematics, suggest research ideas, help edit papers, collaborate with you, offer advice, supply financial support, listen to your thoughts, introduce you to other mathematicians, advocate for you, write letters of recommendation, or cheer you on. Some mentors may be more like friends, others more like a boss—both types of mentors are valuable. Here’s some advice from experienced mathematicians about how they found mentors beyond their thesis advisors.

**From Charlotte Chan (UM PhD 2018, NSF post-doc Princeton, currently Moore Instructor, MIT):**I found it much easier to talk to people once I had results of my own. I could use my experience and intuition in the tiny area I had cut out for myself to join the conversation. The most important lesson I have learned about talking to people is that you don’t always need to follow step-by-step implications in realtime. I consider myself a slow thinker, so I often find that it is more productive to focus on pattern-matching between the conversation at hand and my own research.**From****Harry Bray****(PhD Tufts, 2016; UM postdoc):**I took advantage of or engineered many natural opportunities for potential mentors to approach me. One way I discovered mentors was by going to conferences and giving presentations, at every opportunity (e.g.- poster sessions, lightning talks, parallel sessions). Specifically, I invested time and energy into preparing**accessible**, interesting presentations, which share at least some ideas of the proofs. From these experiences, I was often approached by mathematicians interested in my work who were excited to discuss it or related things, and answer my questions about their own work. I found many fancy mathematicians who care about students, who do not judge, and who would much rather see an accessible talk that their own students would benefit from, than a talk for which they (the fancy mathematician) are the target audience.I emphasize accessible above because from what I can tell, student talks generally don’t solicit this sort of attention. I think mathematicians are always very pleasantly surprised when a student gives a talk that they actually can understand. In my experience, the response to this is positive and has been a starting point for many conversations from which I learned a great deal. I will also insist that presenting well is a learned skill that requires practice, and choosing to invest in this skill may be strategic.I also went to many student-centered workshops, and at these events I tended to ask (sometimes “stupid”) questions shamelessly. The mathematicians who like to speak at these workshops often make good research mentors – especially those who are very happy to answer even small questions, and who are able to be very concrete and accessible in their responses.Some of the people that I interacted with in these moments went on to be my research mentors, and we may not have papers together, but I have learned a lot of math from them and they keep appearing in my acknowledgements sections. One of these people is my older academic brother, who did not overlap with me at graduate school but who has been a wonderful presence in my academic life. Two others write me letters of recommendation. Another one of these people is now a collaborator of mine. He saw a poster I gave and presented a question to me, and we have two papers now with more coming.**From****Oliver Pechenik****(PhD 2016 from UIUC, post-doc at Rutgers, then NSF post-doc at Michigan):**Some of my main collaborators have been my academic siblings. Unsurprisingly, we often have similar mathematical interests! However, we also often have complementary skills and I’ve learned a lot through working with each of them. The rest of my collaborators I’ve mostly found at conferences. At first our interactions were probably me asking bumbling questions about their research and then in turn fumbling through a description of what I was working on. But over time, running into each other at various conferences, we came to understand each other’s mathematical strengths and interests. At that point, it became very easy to transition from casual mathematical chatting into collaboration because we understood who to talk to for help on what sort of issue. Often the main idea of a proof gets worked out during the conference itself and then the details follow via lengthly email chains, video chats, and shared TeX files. The takeaway, I guess, is that it was important to me to attend lots of conferences as a grad student as I tend not to collaborate with people until I’ve gotten to know them fairly well.**Caleb Ashley**(PhD 2013 Howard University, UM post-doc): work with folk who inspire you, and are not a$$holes.**From****Jessica Sidman****(UM PhD 2002, NSF post-doc at UC Berkeley):**Before I gave my first conference talk my advisor told me to think of myself as “a young mathematician, not as a student.” At previous conferences, I had talked mainly to other graduate students and wouldn’t have dreamed of, say, asking a question after talk in public. But at this conference a speaker referenced a result of Sturmfels, and I had a question about it. I asked the speaker, and she said she didn’t know the answer, so I found Sturmfels and asked him. He said one of the senior mathematicians had just asked him the same question — it was a good one. After my talk the next day he asked me if I would apply for an NSF postdoc with him. Being brave and speaking up paid off big time!After I got my job the best advice I got was from Don O’Shea, “maintain your visibility.” Going to conferences and workshops has been really important to me. Meeting people and maintaining relationships has helped me develop mentoring relationships organically. The questions that people ask each other after talks are great ways to get a feeling for whether you click mathematically and personally.**Jeremy Tyson****(UM PhD ’99, currently math Department Chair at UIUC):**The *single* most important mentor for me was my PhD advisor,**Juha Heinonen**. The late 90’s were a really exciting to study analysis at Michigan, where a completely new field—analysis on metric spaces—was blooming. I met most of the major figures in the area as a graduate student, as they made regular trips to Michigan and I made sure to interact with them, at least socially. Many of those people ended up being hugely important mentors for me throughout my career. But not everyone is blessed to have such an extraordinary advisor … I recommend to students and junior researchers to gather your courage and reach out to people. My longest-lasting and closest collaboration began because I was asked to referee a paper of mathematician in my area who was slightly older than me. After the refereeing was finished , I wrote to him just to ask a couple of questions. That turned into an ongoing discussion for a few weeks, after which he invited me to visit for two weeks. That was 20 years ago, and we’ve now written 17 papers together and I’ve spent more time at his institution of Bern than any other except my own. Older students and postdocs can be hugely beneficial. I’ve seen this with my own PhD students: some of the most productive times for them have been when I’ve also had a postdoc. As a graduate student myself, talking math with my advisor’s other PhD student at the time,**Nages**was extremely important. I would encourage students to seek out advisors who take on more than one student at a time, in order to have a built-in support network.**Shanmugalingam**

**From****Will Traves****(UToronto PhD 1998, but spent many semesters at UM; NSERC postdoc at Berkeley):**I want to support the advice that Karen and Jessica give: asking questions is a great way to develop relationships and meet potential mentors and collaborators. Asking questions can be daunting, especially early on in your career. Some starting successful starting points include: asking a question after a talk about a detail, asking about what mental model they use for thinking about a concept, or asking how they got introduced to a topic. I remember being a graduate student and asking a post-doc to send me her papers (these days you can just look them up on the ArXiv). A year later I met her at another conference and explained the work that I’d been doing. She immediately generalized all my results … in real time! Karen Smith became my research advisor and I eventually graduated as her first Ph.D. student. I urge you to look for mentors in a wide range of settings, and not just for research activities! I’ve learned a lot about life from people I’ve met participating in extracurricular activities like sports or hobbies. Of course, the faculty in your department are a resource, but so are faculty outside of your department (or even outside of your institution). Go to MAA conferences; most sections have a one-day meeting each year. Those are also good places to give a talk; your audience’s limited and diverse background makes speaking at such events an interesting challenge. You’ll encounter a wide array of mathematicians, working in more diverse (and representative) types of employment than at AMS or research conferences. Search out folks at UM’s Center for Research on Teaching and Learning. I’ve recently become very involved at a similar program at my own institution and it provides a venue for me to meet with faculty from across the Academy. Finally, don’t neglect to learn from your peers. Many of my friends from graduate school remain friends and mentors to me today.**From Karen Smith (UM PhD ‘93, post-docs at Purdue and MIT):**As a post-doc, I found it easier to engage potential mentors and collaborators by approaching with questions about the things that*they*are interested in, rather than trying to get them interested in my own work. People are often excited to talk about their theorems and expertise. This is a lonely business, so it feels great when someone is curious about one’s work. Giving good talks—accessible, non-technical, informative talks with new results or techniques— is also very helpful for meeting new people to interact with.