Mathematics has often been accused of being a solitary profession, one that a person can practise without talking to anybody else, one that can be done in the head. One can keep one’s mathematical moorings completely to oneself. Like philosophy and realms of higher thought, mathematics can be carried out completely in the mind. Communicating the intricacies of mathematics is extremely difficult.
Paradoxically, though, the same factors that make mathematics solitary, also makes it one of the most social and communal of activities. The content and excitement of mathematics can be shared across several continents, through letters, through telephonic conversations, and of late, email correspondence. Mathematics as a profession allows networking oportunities for sharing of results and ideas that are not present in professions where physical contact and the “real world” are more important.
Sample the Hardy-Ramanujan story. Shrinivasa Ramanujan, a clerk in Madras, wrote a letter to Harold Hardy (of Trinity College, Cambridge) outlining some original results he had obtained in mathematics. His letter smacked at once of diffidence and self-assurance, his results spoke of great mathematical depth as well as lack of good mathematical schooling. Hardy went through Ramanujan’s letter, and saw the spark of genius in Ramanujan. Thus began a fruitful correspondence between the two, that eventually led to Ramanujan getting invited to Trinity College and working with Hardy on original problems.
Paul Erdos, the legendary mathematician, used to hop around the world everywhere, and yet he never lost touch with any of his friends. It was said that his typical letter began with: Let p be an odd prime…
Letters between mathematicians have often focussed not only on the exchange of mathematical content but even on general ideas in mathematics. The Grothendieck-Serre correspondence, for instance, has created new paths in mathematics at a time when the subject was undergoing a radical transformation.
Today, with the presence of instantaneous electronic mail, correspondence and communication in mathematics has assumed new levels of instantaneous. Imagine the kind of correspondence Hardy and Ramanujan carried out. Ramanujan sent Hardy a letter, it took a couple of weeks to reach (at least). Hardy then read it, wrote his reply, and sent it. That again took a couple of weeks to reach. The net result: Ramanujan had to wait for a month (at the very least) to get Hardy’s response to his results.
Today’s Ramanujan-equivalent can send the Hardy-equivalent an email in the daytime, and expect Hardy’s reply the next morning (by making use of the difference in day and night timings).
Email correspondence has provided us with a potent tool with which we can revolutionize mathematical communication? But are we using the tool effectively? Today, the equivalent of Ramanujan can try his/her luck with many a Hardy. But how many of us are willing to be brave and forthcoming, to overcome our diffidence, the way Ramanujan did?
The sense of community is very crucial to the development and fostering of mathematical research (or, for that matter, research in any area). Summer schools, workshops, seminars, are all aimed, among other things, at developing a sense of community and improving international networking. Today, however, we can build and enter communities through individual initiative, much more easily than before.
As an Indian, I say from some experience that Indians are naturally somewhat disadvantaged at building professional networking communities. The problem lies, to a large extent, with the general attitude of servility that has been ingrained into many an Indian through the social system, as well as the lack of practice in presenting and projecting oneself properly. On the other hand, none of these problems are unsurmountable.
Some questions I will look at:
- What is the role and importance of email correspondence (with professors, faculty member and senior individuals) for a mathematics student, particularly at the undergraduate level?
- What is the role and importance of email correspondence (with peers from different educational institutions) for a mathematics student, particularly at the undergraduate level?
With regard to the first point, it is true in the Indian context that the number of centers of excellence for mathematical education at the undergraduate level is very small, and even those that do exist are fairly small places as far as their mathematics department is concerned. Thus, many a mathematics student fails to find guidance in certain areas within his/her institute, and has only books, journals and the Internet to rely on. The student may be unable to pursue areas of his/her personal interest even in summer camps and research programmes, due to the inability to find a guide who specializes in those areas and is free to take the student on. Thus, the student may at many times be compelled to establish communication via email with somebody he/she cannot access more directly.
Another important incentive for establishing email correspondence is that it gives one a foothold in educational institutions where one may later seek admission for study or summer programmes. For instance, after completing my undergraduate studies, I plan to apply for Ph.D. in mathematics to various places in India and in the United States. Having corresponded with professors in some of the universities I am keen on, I feel a greater sense of confidence if what is going on in the institution and what I can expect once I join.
Establishing email correspondence is also good practice for joint work. My email correspondence with Professor Martin Isaacs of the University of Wisconsin-Madison led to a partial solution of the Extensible Automorphisms Problem and also helped me get a better feel of representations and characters. Further, it have me insight into how one usually goes about solving new problems.
Email correspondence can also increase general awareness about certain areas of the subject that are neglected in one’s own institute. It gives the cross-cultual facor. I got important pointers on where to read up groups and subgroups, as well as some subtleties in the subject, through correspondence with Professor Tuval Foguel of Auburn University-, Professor Derek J.S. Robinson, and Professor Jonathan L. Alperin.
Regarding the usefulness of email correspondence with one’s peers in other institutes.
The advantages are quite similar: there is a natural cross-cultural factor, one stays in touch with the way education is proceeding in other institutes. A student studying at another institute may tell one about interesting courses at that institute, and thus help create a new area of interest. Such a student may also be a valuable source to connect to other senior people at the institute.
I haven’t maintained a large amount of correspondence with students in other institutes (perhaps unfortunate). I have had sporadic contact with my Olympiad-time colleague Anand R. Deopurkar, and of late I have also been staying in touch with some people one year senior to me, who are at various Graduate Schools. Just talking to them and knowing the situations in their various schools has been valuable input for me.
The important thing about initiating and managing one’s own email correspondence, though, is not just what it achieves, but what it symbolizes: individual initative taken in the direction one wants to proceed. Rather than limiting oneself to the resources offered by one’s own institute, one actively takes one’s fate in one’s own hands and proceeds to aggressively fulfil one’s own interests.
So how exactly does one go about establishing email correspondence? What are the pitfalls?
I am pretty much a novice in the area, so my observations are still in the process of getting collated.
- Write to a specific person for a specific purpose. There isn’t much point writing to a person just because he/she has won a Fields Medal. Communication with a person should not be done based on the person’s stature, but rather based on what one seeks to get from that person and whether that person is well-equipped to help in that direction.
I have noticed that many people seem to think of writing to outside people as a matter of raising one’s personal prestige, a bit like moving in exalted iintellectual circles. I think this is an inappropriate attitude because it has implicit assumptions of academic stature taking precedence over the utility of correspondence. It is probably a legacy from the era when knowing the high-ups in an intellectual endeavour is what counted for success.
- Give a brief description of why you are writing to that individual person. For instance, if writing to a person on a knotty problem in string theory, you can mention (truthfully) that you have come across this person’s papers or personal webpage in the subject, or that you have heard of his/her work in a course or from some other individual.
This is not meant as an opportunity to give a glowing testimonial to a person whom you probably don’t even know. Glowing praise for a person you don’t know sounds like fawning servitude.
- Give a brief description of the problem and make it very clear what kind of input is sought. Looking at the many attempts I have made at correspondence, the following stands out: in cases where I set forth 1-2 very clear questions and described the problem accurately, the probability of response was much higher.
Often, students who have a whole lot of their own ideas and have not had the opportunity to discuss these ideas with anybody around them or close to them, seek to make full use of email correspondence by waxing eloquent on their ideas. This is usually couterproductive. The average person does not want to hear your new ideas up front. Present him/her with your questions first, let him/her respond, and then follow up by disclosing your ideas. If it is necessary to first describe your idea in order to ask a question, give a small and self-contained description.
- Ask the other person to point you to references for further study and areas where the problem has been previously considered. By saying this, you acknowledge that it is possible that the questions you are asking may already have been answered somewhere, and that you seek guidance in locating the answer. This also shows to the other person that you are motivated to study yourself and are not using him/her as a doubt clearance service.
- If it fits, give a brief explanation of why you were unable to resolve the problem from standard references, and are eager for further guidance.
What happens after the first mail is sent?
If you don’t get a response, do not be disheartened. There could be a lot of reasons:
- The person was on holiday, or on a conference, or travelling, and is not checking mail.
- The person no longer maintains that email address.
- The person missed out your mail.
- The person did not find your mail of much relevance to his/her area of interest and hence forgot about it.
- The person read your mail and will take time out to reply after a few days. While many people respond in a day, it usually takes about 3-4 days.
- The person is mulling over the contents of the mail.
All these are much more likely than what people often conclude:
- This person is too high to answer a lowly creature like me.
- May be that mail was so stupid that the person didn’t even read it.
- May be i shouldn’t disturb people with such silly ideas and questions.
The advisable course of action in case a person does not respond is to just leave it at that. Of course, investigate the content of your mail, see if you have made any mistakes, and try to find out if the person usually responds to mails. It is best not to send a reminder or follow-up mail, because that sounds like you are holdign the other person accountable and accusing him/her. However, you can send him/her another mail after some time on a different or related topic. Do not try to infer conclusions about the other person being too busy to have read your previous mail. Best not to mention it at all, except perhaps as way of introduction (I had written to you earlier on…)
Once you do receive a reply, go through the reply carefully, mull it over, and send the next mail after you have either done a further round of processing on the reply or with a different doubt. Remember in the next mail to acknowledge previous correspondence (by way of introduction) but not make a big show of it. The worst mistake is to expect the other person to still have your previous mails in his/her inbox. Make each piece of correspondence completely self-contained, making no demands on that person’s memory of previous correspondence.
Remember also to keep track of all email correspondence with each person so far.
Email correspondence is a really fruitful way of expanding one’s mathematical boundaries and working for one’s mathematical future. It’s definitely been that way for me!