What Is Research?

May 9, 2010

Survey on math resources on the Internet

Filed under: Culture and society of research,Web structure,Wikis — vipulnaik @ 6:08 pm

I have put up a medium-length survey on the use of math resources on the Internet. The survey is a preliminary one — the results of this will be used in designing more sophisticated surveys. Take the survey here:


You can view the survey results here (but don’t do this before taking the survey!):



January 14, 2010

Math overflow: further notes

Filed under: Culture and society of research,Thinking and research — vipulnaik @ 12:17 am

I mentioned Math Overflow a while back (also see the general backgrounder on math overflow on this blog). At the time, I hadn’t joined Math Overflow or participated in it. I joined a week ago (January 6) and my user profile is here. Below are some of my observations.

Surprising similarity of questions with questions I’ve asked in the past

Looking over the group theory questions, I found that a number of questions I had asked — and taken a long time to get answers to — had been asked on Math Overflow, and had been answered within a few days. The answers given weren’t comprehensive; I added some more information based on my past investigations into the topics, but it was still remarkable that these questions, most of which aren’t well known to most people in the subject, were answered so quickly.

  • The question Are the inner automorphisms the only ones that extend to every overgroup? is a question that I first asked more than five years ago, when still an undergraduate. I struggled with the question and asked a number of group theorists, none of whom were aware of any pasy work on these problems. I later managed to solve the problem for finite groups, and then my adviser discovered, from Avinoam Mann, that the problem was tackled in papers by Paul Schupp (1987) and Martin Pettet (1990), along with the many generalizations that I had come up with (some variants of the problem seem to remain unsolved, and I am working on them). You can see my notes on the problem here and you can also see my blog post about the discovery here.

  • The question When is Aut G abelian? was a question that I had been idly curious about at one point in time. I couldn’t find the answer at the time I raised the question, but stumbled across the papers a few months later by chance (all well before Math Overflow). It’s interesting that the question was so quickly disposed of on Math Overflow. See also my notes on such groups here.

  • The question How can we formalize the naturality of certain characteristic subgroups is a more philosophical question with no real concrete answers, which I’ve considered for a long time too.

  • The question Balancing problem in combinatorics that I had, based on a generalization of an Olympiad question I had seen, turns out to be part of something called rainbow Ramsey theory, as the answer suggests.

Two things stand out: (i) all these questions are questions whose answers are not well-known (the people I asked didn’t know the answers offhand) but are questions that many people do ask (ii) On Math Overflow, they were dealt with quickly.

I think the situation is similar in many other established areas of mathematics — the answers are out there, but they are not well-known, probably because these problems are not “important” in the sense of being parts of bigger results. But they are questions that may naturally occur as minor curiosities to people studying the subject. These curiosities may either go unanswered or may get answered — but the answers do not spread to the level of becoming folk knowledge or standard textbook knowledge, because they aren’t foundational for any (as yet discovered) bigger results.

Math Overflow now provides a place to store such questions and their answers — thus, next time you have one of these questions, a bit of searching on Math Overflow would yield the question and the answers provided and stored for posterity. Apart from the questions I had thought of, consider this one that somebody thought up and turned out to have been considered in multiple papers: When is A isomorphic to A^3?.

The situation is probably different for areas where new, cutting-edge questions are being asked — i.e., areas where the questions are charting “new” territory and are helping build the understanding of participants. Some people told me that this is likely to be the situation with areas such as topological quantum field theory or higher category theory.

Skills needed and developed

So what skills are needed to participate in Math Overflow, and what skills get developed? In order to answer questions quickly, a combination of good background knowledge of mathematics and the ability to search ArXiV, Mathscinet, JSTOR, Google Scholar, and other resources seems necessary. In order to ask questions, it seems that a combination of a lot of background knowledge and a natural curiosity — of the bent needed for research, is what is needed.

Pros and cons of posting on Math Overflow

The obvious pro seems to be that a lot of people read your question. The sheer number of readers compensates for the fact that most people, even experts in the area, may not immediately know the answer. Because of the larger number of people, it is likely that at least a few will either have come across something similar or will be able to locate something similar or will be able to solve the question because he/she gets curious about it. It was pretty exhilerating when a minor question that I didn’t feel equipped or curious enough to struggle with was answered by somebody who came up with a construction in a few hours (see collection of subsets closed under union and intersection), or when a question I had about elementary equivalence of symmetric groups was answered within a few days (see elementary equivalence of infinitary symmetric groups).

The potential con could be that people may be tempted to ask a question on Math Overflow without thinking too much about it. This probably does happen but I don’t think it is a major problem. First, the general quality of participants is quite high, so even if people ask questions without thinking a lot about them, chances are there is something interesting and nontrivial about the question — because if there weren’t, people of the profile contributing would have been able to solve it even without a lot of thought. Further, even if a question is a good exercise for a person specializing in that subject — so he/she should struggle with it rather than ask others, it may be good for a specialist in another subject to simply ask.

The voting (up/down) system and the system of closing questions that are either duplicates or not suitable for Math Overflow for other reasons (such as undergraduate homework problem), combined with a reputation system for users linked to the votes they receive, seems to be a good way of maintaining the quality of the questions.

A revision of some of my earlier thoughts

In a blog post almost a year ago titled On new modes of mathematical collaboration, I had expressed some concern regarding the potential conflict between the community and activity that is needed to have frequently updated, regularly visited content, and the idea of a steady, time-independent knowledge base that could be used as a reference. mathematics blogs, with regular postings and comments, and polymath projects, which involve collaborative mathematical problem-solving, are examples of the former. Mathematics references such as Mathworld, Planetmath, the mathematics parts of Wikipedia, The Springer Encyclopedia of mathematics, ncatlab, Tricki, and the Subject Wikis (my idea) are some examples of the latter, to varying degrees and in different ways. The former generate more activity, the latter seem to have greater potential for timeless content.

Math Overflow has the features of both, which is what makes it interesting. It has a lot of activity — new questions posted daily, new answers posted to questions, and so on. The activity, combined with the points system for reputation, can be addictive. But at the same time, a good system of classification and search, along with a wide participatory net, makes it a useful reference. I’m inclined to think of its reference value as greater than what I thought of at first, largely because of the significant overlap in questions that different people have, as I anecdotally describe above.

Math Overflow stores both mathematical data — questions and their answers, as well as metadata — what kind of questions do people like to ask, what kind of answers are considered good enough for open-ended questions, how quickly people arrive at answers, what kind of questions are popular, how do mathematicians think about problems, etc. This metadata has its own value, and the reason Math Overflow is able to successfully collate such metadata is because it has managed to attract high-quality participants as well as get a number of participants who are very regular contributors. Both the data and the metadata could be useful to future researchers, teachers, and people designing resources whether of the community participation type or the reference type.

On the other hand…

On the other hand, Math Overflow is not the answer to all problems, particularly the ones for which it was not built. Currently, answers to questions on Math Overflow are broadly of the following three types (or mixtures thereof): (i) an actual solution or outline of a solution, when it is either a short solution arrived at by the person posting or an otherwise well-known short answer (ii) a link to one or more pages in a short online reference (such as an entry on Wikipedia or any of the other reference resources mentioned above) (iii) a link or reference to papers that address that or similar questions.

For some questions, the links go to blog posts or other Math Overflow discussions, which can be thought of as somewhere in between (ii) and (iii).

With (i), the answer is clearly and directly presented, and the potential downside is that that short answer may not provide a general framework or context of related results and terminology. With (ii), a little hunting and reconstruction may need to be done to answer the question as originally posed, but the reference resource (if a nice one) gives a lot of related useful information. (iii) alone (i.e., without being supplemented by (i) or (ii)) is, in some sense, the “worst”, in that reading a paper (particularly if it is an original research paper in a journal) may take a lot of investment for a simple question.

In my ideal world, the answer would be either (i) + (ii), or (ii) (with one of the links in (ii) directly answering the question), plus (iii) for additional reference and in-depth reading. But there is a general paucity of the kind of in-depth material in online reference resources that would make the answer to typical Math Overflow questions by adequately dealt with by pointing to online references. So, I do think that an improvement in online reference resources can complement Math Overflow by providing more linkable and quickly readable material in answer to the kinds of questions asked.

October 26, 2009

Polymath again

Filed under: Culture and society of research — vipulnaik @ 10:42 pm

Timothy Gowers and Michael Nielsen have written an article for Nature magazine about the polymath project (I blogged about this here and here).

In the meantime, Terence Tao started a polymath blog here, where he initiated four discussion threads (1, 2, 3 and 4) on deterministic ways to find primes (I’m not quite sure how that’s proceeding — the last post was on August 28, 2009). (UPDATE: A new post (thread 5) was put up shortly after I published my blog post).

Around the same time, Gil Kalai started a polymath on the polynomial Hirsch conjecture (1, 2, 3, 4 and 5).

Also, some general discussion posts on polymath projects: by Tim Gowers and by Terence Tao.

It remains to be seen whether any of these projects are able to reach successful conclusions or make substantial inroads into the problem. If there is another success for a polymath project, then that would be a major booster to the idea of polymath projects. Otherwise, it might raise the question of whether the unexpected degree of success of the first polymath project led by Gowers (which aimed for, and got, an elementary proof of the density Hales-Jewett theorem) was an anomaly.

March 17, 2009

The “fair copyright in research works” controversy

Filed under: Culture and society of research — vipulnaik @ 4:38 pm
Tags: , ,

There are competing interests, and there are competing interests. In recent years, there has been a growing Open Access (OA) movement, that advocates that scientific research be published for open access. So, not only should people be able to download scientific papers for free, they should also be able to reuse the results for their own experiments. Creative Commons, an organization that came up the the Creative Commons licenses that allow for creators of original works to give others rights that go beyond fair use in copyright law, has been among the organizations at the forefront of Open Access. They even have a separate branch, called Science Commons, that specifically deals with opening up scientific research.

Science Commons was understandably delighted at the NIH Public Access Policy. According to this policy, the National Institute of Health (a governmental organization of the United States) mandated that all research conducted with NIH grants be released to the public within a year of publication.

Recently, a bill was proposed by John Conyers, a Democratic House Representative from Michigan (a United States state) that would have the effect of scrapping the NIH public access policy, and effectively making it impossible for such policies to be instituted. The act was titled the Fair Copyright in Research Works Act”. Many in the blogosphere and scientific community were up in arms at this act. Consider, for instance, that 33 Nobel Laureates apparently opposed the bill. Or consider two posts by Lawrence Lessig and Michael Eisen in the Huffington Post, suggesting that Conyers’ decision to reintroduce the bill was influenced by the campaign contributions he received from commercial publishers. Conyers’ reply was severely trashed by Michael Eisen, Peter Subers, and Lawrence Lessig. Blog posts by Paul Courant, Molly Kleinman, and numerous others. Clearly, the bill hasn’t gone down well with a lot of open access advocates.

What precisely is the problem?

There are a number of arguments for making open access mandatory rather than voluntary, at least in certain contexts. One is the moral argument: taxpayers are paying for the research, so they have a right to access that research. This argument in and of itself isn’t extremely strong. After all, if we really want to be fair to taxpayers, shouldn’t taxpayers get to vote on how the NIH spends its money in the first place. Also, considering that only U.S. taxpayers are paying for the research, shouldn’t the open access policy make the materials openly available only in the United States, so that other countries do not free-ride off the money of U.S. taxpayers? Further, shouldn’t people who pay more taxes be given greater privilege in terms of the level of access they get? Further, considering that most taxpayers would not be interested in reading medical and health-related research, doesn’t this effectively subsidize researchers at the cost of publishers, while keeping the majority of taxpayers exactly where they are?

The taxpayer argument doesn’t seem to be a very strong argument logically, but I think it carries good moral and emotional overtones, which is why it is touted so much.

A more valid reason is the argument that opening up the findings of research creates very strong positive externalities for further research. Thus, a lot of other researchers can benefit from opened up research and thus benefit science, even though these researchers would not have found it personally worth it to subscribe to the journal. Further, science papers, once freely available, can be used by people who may not have tried them out in the past, such as high school students, people with health problems, and others. (more…)

December 21, 2006

Culture and diversity in mathematics

Filed under: Culture and society of research — vipulnaik @ 9:34 am

Recently, in India, there was great furore about the introduction of an additional quota for Other Backward Classes for reservations. These protests opened up a whole can of worms about why educational and professional institutions do not have proportional representation of different parts of the population. The pro-reservationist argument that if 20% of the populace of the country belongs to a certain social class, so must 20% of the populace of its educational institutions, rings true for its very simplicity.

Reservation actually reminds one of the closely related notion of Affirmative action. The aim of affirmative action is to positively discriminate towards groups or communities that have been discriminated against in the past. The obvious goal of affirmative action, of course, is justice, justice to communities and groups that have not had a voice so far. Another goal of affirmative action is discovering talents that would have been suppressed due to discrimination. The most interesting claim made by proponents of affirmative action, though, is that affirmative action promotes cultural diversity. The arguments for this are so often quoted that they sound trite: different cultures give rise to different perspectives, different ways of approaching a situation, and hence cultural diversity is important in any area. Hence, Proportional representation of all groups is the desired goal of any public institution, whether academic, professional, social or political.

If the argument sounds specious in this extreme form, we can still ask: to what extent is it true? How does cultural background influence the way one approaches a mathematical problem? Does being Asian, American, or African, endow one with particular ways of thinking that help in the solution of mathematical problems. In other words, how does generic culture influence mathematical culture?

First, what are the ways in which background can influence mathematical ability?

  • Genetic factors: Even though it may be politically incorrect to argue that genes could be responsible for mathematical superiority, the contribution of genes can definitely not be ruled out on a scientific basis.
  • Toddler education: The way a child perceives numbers and shapes is often shaped in his/her initial years. The quality and nature of mathematical education in these years can profoundly influence the child’s conception of numbers. Also, the practical situations in which he/she may use numbers at an early age can also play a profound role in the development of the child’s intuition.
  • Primary and secondary education: The nature of the curriculum, the course pressures, and the opportunity to express and develop talent in mathematics are definitely importnat determinants of how the child’s mathematical talents are shaped.

The argument for cultural diversity in mathematical institutions can gain strength if we show that different cultures are strong in these respects in different ways and to different degrees. That is, we need to argue something like: while a person from culture A is naturally better equipped for manipulating algebraic symbols, a person from culture B has a better feel for geometric figures. The important point, though, is that the arguments must actually be substantiated by hard facts. Let me look at a very clear and very definite cultural difference that is famous in the history of mathematics: Shrinivasa Ramanujan.

Shrinivas Ramanujan spent his childhood in Sarangapani Sannidhi Street, Kumbakonam, Tamil Nadu, born to a particular sub-caste of the Brahmins. Taking a simplified view, the Brahmins are the highest in a four-tier caste system in Hindu society, and the professed purpose of Brahmins is sacred learning. Thus, naturally, being from a Brahmin background meant that Ramanujan has the support and backing of his family (this is all in relative terms to the times) to play around with numbers. It also meant that others were willing to give him a more patient hearing and more of a chance to pursue and study mathematics than they would have given somebdy from a lower caste. In fact, it is inconceivable that a Ramanujan would have emerged from a non-Brahmin family at the time.

Thus, one cultural factor that played in Ramanujan’s favour was the fact that he was born a Brahmin.

What makes Ramanujan stand out is not just that he was a great mathematician, but that he had a whole lot of intuition and insight into mathematical truths, and could make astounding statements in mathematics even without formal proof. This has been partially explained by the mystic traditions in Hindu society. In Hindu society, the most rigourous of science developed alongside superstition and mythology, so much so that the two were completely intertwined and difficult to separate. The need for proof or rigour never arose; all results were justified by divine intervention. In this milieu, it was natural that Ramanujan tried to see the God directly rather than give conclusive and demonstrative proof.

This contrasts sharply with Europe, where the development of mathematics built on the foundations laid by the Greeks. Even though arguments by the Greeks were often flawed, they laid down the rules for logic and attempted to stick to those rules, very rarely seeking divine inspiration.

Thus, Ramanujan, as the intuitive mystic we know him to be, may not have existed had he been born in Europe. His talents may have led him to become a mathematician, even a first-rate mathematician, but the lure of the rigour of proof may have slowed down the steady stream of intuition which he poured out.

Is it?

It is really hard to say what Ramanujan could have achieved had he been born into a world where mathematical rigour was valued from Day One. It is possible that within this environment, he would have been able to combine his skills of grasping the results directly with the ability to give conclusive proofs, and gone much, much further. But there remains the possibility that he would have gotten himself restricted to a much more mediocre (though possibly happier) life.

Still, we do have one example of how a person’s culture is likely to have influenced his approach to mathematics.

However, the example of Ramanujan hardly gives grounds for the introduction of cultural diversity in institutions. Considering that the Indian culture could produce hardly any other mathematician, the fact that Ramanujan was produced by it seems scant justification for introducing a quota for the Indian perspective into mathematics.

In fact, what I feel is that mathematical culture is a very different ballgame from social culture. While social culture may have a few influences on mathematical culture, the fact is that mathematical freedom, creativity, expression and modes of thinking may often starkly contrast the level of freedom, creativity and expression and the modes of thinking existent in general society.

China, Japan, and states in the erstwhile Soviet Union, have produced a whole lot of mathematicians. So have countries like Hungary and Romania. What lies behind their success? A bit of genetics, a sound primary schooling system, and the development of a strong mathematical culture.

Hungary has a rich tradition of mathematics largely because it gave rise to mathematics greats like Paul Erdos, Bela Bollobas and so on. These people were not only a source of inspiration to others but also led to the creation of an atmosphere of mathematical contests in the country meant to sieve and train mathematical talent. The contests, the journals, the puzzles, all made people look forward to doing mathematics, they helped provide a meaning to mathematics.

A country like Russia, with its tremendous emphasis on a problem-solving culture, has naturally been producing great mathematicians. China, with its procedures for selecting and identifying talents at a young age and training them, has consistently been performing well at the Olympiads.

The United States has naturally benefitted and recognized talent and pushed it forward. But as such, the United States has very little of mathematical culture to speak of. It has no zeroes, no number systems, no pi’s and no right triangles to its credit. Yet, it has forged to be a leader in mathematics, with its educational institutions being the much-sought destination. Why? Because its educational institutions, at all levels, have sought to create their own mathematical culture. Each individual has brought in personal initiative and ideas towards fashioning a culture.

Arnold Ross kickstarted mathematics education for young kids in the United States, a precursor to United States’ current Olympiad programmes.

India is culturally poor in mathematics, because we Indians have been unable to create a culture for the kind of mathematics that is practised today. We cling to the importance of Vedic mathematics, spouting its cultural role, forgetting the fact that its primary role is only as a more effective means of computation — a goal that is itself outdated. We talk of the mystic element and the natural intuition of Indians in mathematics, with only Ramanujan to boast of, and he too being a moot point. Our ministers make misguided attempts to revive Indian culture by teaching us to treasure the zero and our numeral system and Vedic mathematics and linking all of them to mysticism and tradition.

And yet, the progress that India as a country has been seeing in creating mathematics and mathematicians is due to the mathematical culture created by individuals. We have not achieved cultural diversity, but at least we are on our way to getting a culture. Home Bhabha started off TIFR, a leading institute for mathematical research, where a culture for mathematical thought was developed. Alladi Sitaram, with others, started the Institute of Mathematical Sciences in Chennai, and this further fostered the mathematical culture that has been growing in South India since the time of Ramanujan. Seshadri started Chennai Mathematical Institute to propagate mathematical culture to students right from the undergraduate level.

Cultural heritage? Yes. But not the culture of tradition, of ancient values. It is not the invention of the zero 1200 years ago that moves a culture forward, it is the existence of a system for educating, harnessing and nurturing talent that counts. The past is just a stepping stone to the present.

If we seek cultural diversity within mathematical institutions, then the cultural diversity we need to seek is diversity of mathematical culture. The first prerequisite, though, for such diversity is that there should exist a mathematical culture in the first place. And to have this mathematical culture, there first of all needs to be an identification and training of mathematical talent.

What deep insights can an illiterate Indian give into mathematics by virtue of being born into a different culture? How does an African, who has no home support for the pursuit of mathematics and has struggled to study from textbooks publised in America, provide a unique cultural perspective to mathematics. How does a specialist hotelier bring new ideas to the study of differential geometry.

Cultural diversity in a community develops once culture gets developed. When the culture is well-set, trends start getting formulated. For instance, suppose a center for mathematics is established in a city, and young students come to that center to discuss and solve mathematical problems. Then, the cultural direction of that center may depend on who decides the collection of library books to be stocked in the center, who conducts classes and gives lectures there, and what policies the center follows. If the center decides to promote mathematical contests with a particular flavour of problems, a culture for those problems is likely to develop. If the center decides to recruit people and conduct lectures in a particular area, a culture for that area is likely to develop.

The mathematical culture itself derives from the mathematical institution rather than from any bare-bones culture. It isn’t Chennai that has a mathematics culture, but institutions in Chennai, be they Chennai Mathematical Institute or the Institute of Mathematical Sciences or the Association of Mathematics Teachers of India. It isn’t Cambridge, Massachussetts that has the mathematics culture but MIT and Harvard. A culture rarely comes imported from the past, the culture is created by individuals and institutions, and only then permeates to the city and the community.

Instead of seeking diversity for our mathematical institutions along the lines of social and economic diversity and diversity of generic culture, we need to seek diversity of mathematical culture. But, in the Indian situation, there hardly exists a mathematical culture in the first place. It is thus necessary to first create a mathematical culture, then let it diversify into many cultures, and then automatically reap the benefits of the diversity.

Mail correspondence

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!

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