Getting Started in Research Frank Sottile How do you get started on research, post-Ph.D.? That is, how do you make the transition from graduate student to a budding research mathematician? Specifically, how do you go from working on a problem for several years with their advisor's attention and counsel, to independently seeking out and solving your own problems? Anecdotal evidence and advice I have been given point to this being a critical transition, one which many (most?) young mathematicians do not make. I don't really have the answer. Mathematical research is a very personal pursuit, and what works for one may not work for another. However, I think I have made that transition, and I have thought a lot about it and have some observations, which I share here. I will refer to events in my career, and so this won't apply to everyone, but hopefully it will either stimulate discussion, or help someone. I managed to prove a new result within two months of starting my first job. I feel that I was fortunate to get so quickly started in new directions, but also that it was absolutely necessary; you can't go very far extending your thesis or living off ideas that your advisor sends your way. The key is to generate your own ideas. This may seem rather daunting. One way to do this is to attend lots of talks and conferences and talk mathematics with as many different people as possible. I also recommend casting a wide net for potential problems; not only are you more likely to find something to your liking, but you will learn more along the way. You don't have to work on problems similar to your thesis for the rest of your career. That said, you should publish the main results from your thesis. I didn't do this for about a year, and the delay later cost me a lot of time. Instead, I moved to a new city and got settled in. I had heard of people who moved, got settled in, then their first term started and they got caught up in their new responsibilities. After a couple of terms of that, they hadn't done any research and found it difficult to get started. To avoid this, as soon as I arrived in my new city, I started to think about two ideas I had earlier in my graduate career, but unrelated to my thesis. They were vague, and I didn't get too far on them. One was cut short by the appearance of a manuscript - by someone I had innocently talked to the previous spring. I still think the other is interesting, but school quickly started, and by the time I had the time again, I had other ideas. The first week of any academic term is a week to organize your classes and schedule and see your colleagues after a break. It doesn't seem to be a good time to try to get any research done. If this is a new job, then there is a lot more dislocation: adjusting to new responsibilities, a new town, new department, and new people. It took me several weeks before I was able to do much else, and by then I had other responsibilities: I was scheduled to give a seminar at my home institution, and then speak at a regional conference the next weekend. Giving and attending talks is an important component of my research activity. I use talks, particularly at my home institution, to help me organize my ideas or cast them in a new light. It is a challenge to present a coherent explanation of your work in any format. The choices of what and how to present the material is always leads to improvements in how I think about them, which shortens proofs and suggests new avenues to explore. Giving a talk introduces you to your audience, and such introductions lead to worthwhile interactions. This is also a reason to attend seminars, colloquiua and to travel to conferences. I recommend footing the bill for some travel yourself if no outside support is available. After all, this is a vital part of the mathematical enterprise. Another responsibility was a grant application, which turned out to be one of the better uses of my time. In a grant application, you must convince other mathematicians that your good ideas are worth supporting. The exercise of writing one forces you to do overcome a big post-Ph.D. stumbling block: generate new ideas of what to work on. Along with introspectively trying to describe what I was interested in, its value, and what I might want to do next, I contacted several people at other institutions and started a dialogue. I have recently written two joint papers with people I first contacted at that time, almost 30 months ago. The value of those contacts isn't these recent papers, but how they helped my thinking about mathematics to evolve. How do you make contacts with others when you are a fresh Ph.D? I already mentioned giving talks and attending conferences. When I travel, or when the department has visitors, I socialize and ask people questions that come to mind. It is easiest to get to know people who live nearby. I knew one of my coauthors from his papers, and met him when I learned that he had also just moved to the same town. One way I have met other mathematicians is by distributing copies of my papers. When I complete a manuscript and get ready to send it to a journal, I also make a list of other mathematicians who may be interested in these results, and then I send them copies with a cover letter describing the main results and sometimes pointing them to parts of the paper which may interest them. This list is generated by people whose papers I have read or cited, people who I have heard speak or heard about, people who have asked me for copies of my work, and my local colleagues. I am not shy about this; at worst, they won't read my paper. I also put my papers on electronic preprint servers, and plan to eventually post them on a web page. I have met several people this way, some of whom have had me visit or give a talk and one with whom I just wrote a paper. When writing that grant application, one of the people I contacted pointed me to a conjecture (in algebra) in one of her papers that I added to my mental list of problems worth thinking about. After finishing the grant application, I hadn't done any new work in the several months since I handed in my dissertation. Rather than get started, I traveled to the Midwest to give a seminar, visit a friend, and attend a weekend conference. While there, I had a flash of insight one morning in the shower (really!). I realized how it should be possible to prove the afore-mentioned conjecture using geometry. Generating ideas is necessary, but then you need to bring them to fruition. Research, like other quality pursuits, requires time and dedication. Discipline may be the most important part of my doing mathematics. I impose deadlines on myself, reserve time solely for research, and try to use my time efficiently. In short, hard, dedicated work is essential. I jealously guard my research time; otherwise it evaporates as there are many other demands on my time. When I returned home after that trip. I arranged my schedule to give myself some uninterrupted blocks of time. This was a matter of changing when I did my class preparation and other necessary tasks (email, meeting with my TA's, grading homework, lunch, etc.) so that I would have these periods. Then I used this time to try and prove this conjecture. After a very dedicated few weeks, I did it. After a couple of hours of euphoria, I began the hard part - writing up this result and obvious extensions. I am not kidding, I find that I spend more time writing than research, and it is not nearly as fun. It is absolutely necessary to do so - if my papers are not carefully written and if I do not make every effort to improve the exposition and clean up the arguments, then few will read my papers. Remember, your readership is a monotone decreasing function of line number. While writing this result up, I started to work with a coauthor on a further extension, which would have been a fantastic result. After six months of my doing nothing else, we gave up. This failure to get results is quite common, a lot of promising lines of my research fizzle out, or else I cannot solve them. Don't let this keep you from trying. When working on a problem, you inevitably learn some new mathematics or get some ideas, or obtain partial results. Partial results are not necessarily second-class; the problem might be harder than you thought. If the partial results are interesting and lead somewhere, then they are publishable. Only last summer, my coauthor and I came back to this work and write a paper whose genesis was the successes we had among our many failures. I now think this paper is the best I have written, and we have several more in planning. While we have yet to solve the problem we set out to do, this work has my most interesting line of research. During the last half of that first year, I did other things, as well; I gave several talks and attended an instructional conference in an area that I am only now just beginning to work in. There I met someone who was very enthusiastic about my thesis and gave me the encouragement I needed to write it up, and who has now become a very important mentor. I also met a student there whose research gave me an idea of an application of some geometric constructions I had been thinking about. This helped me complete those constructions and write them up, and now we are writing a joint paper on those applications. At the end of my first year out, while I had failed even to massage my thesis into a preprint and had only solved one new problem, I think that I was on the right track. I had made serious work a regular habit, I was in regular contact with a number of other mathematicians, and I was thinking about a number of different problems.