A common question that I get is, what is mathematics research like? Or other variations of this such as, what do you do as a postdoc, teach or take classes? Or, oh you’re a postdoc, so which degree is this for? Or, oh, so you want to be a teacher? So let me fill you in as to what exactly I did with most of my time here as a postdoc at Hebrew University.
While many postdoctoral positions in the US have some minimal teaching responsibilities (one to three courses per year), the postdocs here (or at least the foreign ones) have no teaching responsibilities. This means that for the first time in my life I had the luxury of spending all of my working hours on research. (My PhD granting institution only gave stipends for teaching except for some rare exceptions.) It was as if a whole new world had opened up to me. What did I do with my time? Most of it I spent reading various articles and books. In some cases, I read material relating to areas of mathematics where I felt that I needed to strengthen my background, while in others, I read because I’m drawn to the topics and want to actively contribute to the field. After several days of reading, I would often have a list of questions that I wanted to answer. Most of these questions were “basic” in the sense that I needed to answer them in order to feel like I understood what I read. Some of the questions, however, turned into research problems that I have either turned into papers, I am currently working on them or I am planning on addressing them in the near future. Answering these questions involved some combination of devising a strategy of proof, more reading, working out specific details of the strategy, consulting with other mathematicians (especially those that are more senior to me) and possibly modifying the strategy or the question and then, eventually, after several iterations, a complete answer to my question (or a variation of my original question). You might ask why modifying the question be appropriate? Well, when trying to prove something new, sometimes we have to settle for something less or different than our original goal. For example, the question I wanted to investigate for my Fulbright project was whether or not certain types of universal deformation rings are unobstructed for all but finitely many primes. (The conjecture was that these rings are unobstructed for all but finitely many primes.) After spending several weeks on this project, however, I realized that in general there are obstructions coming from lifting the determinant of the residual representation of interest. I then spent several more weeks trying to get a precise handle on when exactly these obstructions occurred and so on. Meanwhile, I simultaneously forged ahead on other fronts of the project, exploring other aspects of my original strategy. Happily, every other part of my strategy seemed to be falling into place. This left me confused as to what I should be trying to prove or where exactly I should take this project. Eventually, I consulted with my postdoctoral adviser about the progress that I had made. He told me that since everything else was working out so beautifully, I should get around this issue of lifting the determinant by looking at universal deformation rings with fixed determinant instead of the original problem that I had proposed. In hindsight, this was the obvious answer, but I feel very fortunate that he kept me from spinning my wheels for too long on this issue.
Let me just stress again, that most of these questions were what are known as “exercises” and so usually consisted of just unwinding definitions or using a definition plus one or two bigger theorems. In these cases, I didn’t have to go through the whole process I just described. It just took me a matter of anywhere between 5-10 minutes or perhaps 2 – 3 hours to work out what I didn’t understand.
The other part of research that I have yet to mention is attending seminars, classes and conferences. All three play a role in expanding our knowledge base, exposing us to new ideas, giving us inspiration for our own work and providing a venue for networking and collaboration. Hebrew University ran numerous research seminars almost every week. During these seminars researchers from around the world would come to present their work. I often went to at least one of these per week and sometimes as many as 4 or 5. The department also offered several high level courses or mini-courses on topics for which no standard text has been written or which are of current research interest. I also attended some of these courses and tried to learn a few new tricks and see how my research related to them. Finally, one of the nice things about being at a research hub is that conferences are often held at your home university. This was definitely the case with Hebrew University and I definitely took advantage of this by attending parts of about seven different conferences at the university. I also traveled to participate (speak at and attend) another three conferences. I often find that the most productive part of attending these conferences is the opportunity to talk with other mathematicians informally. By doing this, we can test out ideas for new projects, get advice from experts and form relationships that might potentially lead to a collaborative project down the road or perhaps just a friendly ear when we’re faced with a difficult situation.
Let me conclude by addressing the questions that I posed at the beginning more directly. While I occasionally attended selected classes, I did not teach any classes during this postdoc. A postdoctoral position is just that, a first job after finishing a Ph.D. (according to academic standards – which differs from the ministry of the interior who calls me a student). Yes, I do want to teach. In fact, I really enjoy sharing this passion that I have for mathematics with my students. I also want to continue to be an active researcher. I hope that one day I will have tenure at a college or university where I don’t feel like I have to sacrifice one for the other. Finally, I hope I’ve at least given you a glimpse of what I did with a large chunk of my time here and what it’s like to do mathematics research.
The following are some photos of me in my “natural” habitat taken my dad on his visit in February of 2013.
In “my” office that I shared with 4 others
A mathematical quilt
At the entrance to the Manchester bulding (i.e., the math department)
Only a few more days (and posts) left on this adventure…
Adam
Adam and Alli's Fulbright adventure in Israel 