Research in Mathematics

Working in mathematics, I’ve found myself often asked the question “What do you do?” Sometimes the expected response is my “elevator pitch” (the short blurb about my area of expertise). But sometimes the question is more basic: “What is it you do, though? Do you just sit all day and think?”

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Now, to a large extent, many people in research spend all day thinking. However, mathematics is not simply the art of staring at a problem until the solution materializes in one’s head. (It’s worth a try, but often solutions do not come from epiphany alone.) I would like to discuss a few of the ways in which research is conducted in mathematics, with emphasis on the parallels and similarities that may exist between mathematics and other fields, perhaps to somewhat debunk that notion there may not be any such similarities.

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Mathematics research revolves around proving new theorems — mathematical statements that can be deduced from the fundamental axioms of mathematics and from preexisting theorems. Generally, though, the procedure is not to make a big pile of the existing statements and to try to string them together randomly until one forms a coherent deduction that results in something meaningful. That would be pretty rough sailing! Mathematics relies on conjectures, put forth as believed to be true and hopefully proven by someone at some later time. While there are conjectures (e.g. Goldbach’s conjecture) which remain unsolved for long periods of time (sometimes resulting in notoriety), most theorems start out as rough ideas or propositions that are developed with increasing structure and refinement until they are proven. In addition to proving new theorems, other steps forward in research include constructing examples of mathematical structures and verifying theorems by re-proving them in new ways. Computational work is also done to improve theorems in the case that a theorem is quantitative (or sometimes, to prove that a quantitative result is best-possible).

El. J. Comb. logo

Know the literature: As in most fields, the mathematical literature is vast, and perhaps especially in mathematics, it is easily accessible. Increasingly, mathematics journals are available online — not just through the library system, but free for instant download on the Web. Having less concern for the preservation of intellectual property, many editorial boards have shifted to such open/free publication. (Indeed, I myself have a publication in The Electronic Journal of Combinatorics, which is precisely such a journal.) There is also the arXiv (the X is pronounced sort of like χ, the Greek chi), which hosts preprints of papers and other works of mathematics (and many other fields).

Being familiar with the body of literature, both seminal papers and other older works as well as the current cutting-edge work (as it appears first, usually on the arXiv), is an important part of conducting research in mathematics. Jacob Fox came to Rutgers in 2009, when he was at Princeton, to speak at a seminar. He noted during his talk the importance of being familiar with the literature, mentioning in particular how his knowledge of a certain publication helped him and his coauthors solve a problem.

Mathematics [cc]

Crafting and Proving Good Conjectures: One of the more important questions is where to start — if we’re going to prove a statement is true, what is that statement? Generating good conjectures is not a matter of guesswork or divine inspiration, at least not entirely (although the former may have helped from time to time, and the latter is open to some debate at least). Increasingly, experimentation is a common way to generate conjectures. It is also often useful to test conjectures in small, typical, or special cases (where “small,” “typical,” and “special” depend on the problem at hand). Usually a conjecture applies to too many cases to test them all (sometimes, infinitely many cases), so this methodology is often used to verify that the conjecture is sometimes true, but not to verify the conjecture exhaustively. (Conversely, experimentation may lead to a disproof of a conjecture by identifying, constructing, or otherwise elucidating a counterexample.) Experimentation may also help unearth components of the proof of the conjecture at hand.

It is also crucial to have a firm understanding of the big picture in the field where these questions are being asked. There is a substantial amount of context and content that guides someone to the right kinds of conjectures and the proofs of those conjectures. Mathematics is a field in which the objects of study are highly structured, and knowing these structures helps eliminate some of the technical clutter that can obfuscate the underlying truths that one wishes to prove and the bits and pieces that go into proving them. Many proof techniques can be adapted to different situations, so in some sense theorems may be proved by matching a generalized proof to a statement you would like to prove specifically.

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Building Theories and Solving Problems: Tim Gowers is famously credited for roughly dividing mathematicians into the two categories problem-solvers and theory-builders (or rather, he is credited for noting this division in his oft-quoted The Two Cultures of Mathematics). I won’t discuss this dichotomy, but these two activities characterize much of the research done in mathematics. Proving single, unrelated theorems one-by-one is not usually how research goes. Rather, the enterprise involves longer strands of investigation — a dozen theorems sometimes collapses into a single stronger and better statement after enough exploration and refinement. Meanwhile, single ideas branch into many avenues of investigation. But generally, the aim is not to knock down one theorem, then turn around π radians and start over, but to work on larger-scale investigations. I could make a metaphor about bowling pins or dominoes, but I think the idea is clear. An important aspect here is also collaboration, which is a major element of research for many mathematicians. Working on papers is one part of collaboration, but other important activities include seminars & conferences (as participants and as organizers), expository writing, editorial work, and many other collaborative activities.

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So the venture is to find good lines of research and establish some clear, path along that line. There are the two approaches. The first is to identify important problems and build up theory to solve them. One famous example is Fermat’s last theorem, which conjectured centuries ago and recently proved by Andrew Wiles. During those centuries, large swaths of mathematics were developed in large part as attempts to prove this conjecture (including Schur’s theorem, one of my personal favorites). This “problem solver” work weaves what might be the leading strands of the theory, loose and rough but pushing outwards farther than neighboring strands. Such work often moves mathematics in innovative or interdisciplinary directions, building bridges between fields of mathematics, and may also connect with work in applied mathematics. The “theory builders” weave strength and cohesion into the fabric (to extend the metaphor). To this end, they focus their research on developing and enriching the theory. They may work to classify all types of a particular structure, for example. Such work includes that of several Rutgers faculty in classifying the finite simple groups. This theory-building reinforces others’ work as they develop and solve conjectures, as it makes the underlying theory more robust.

Images used in this entry are used under fair-use and/or under licensing guidelines set forth by the copyright holder that allow use in this blog, as presented for educational or critical commentary. Images are copyright their respective holders and credit or source is indicated in each caption or in the text of this entry, as applicable. Thanks to Yusra Naqvi for her helpful comments and suggestions.

Author: Kellen Myers

I am a Mathematics graduate student at Rutgers University. I am a Ph.D. candidate, which means at this point, my work is towards writing my dissertation. My advisor is Doron Zeilberger. My research interests are in combinatorics. A few specific interests are within Ramsey Theory, especially Ramsey Theory on the integers. This means I have an interest in both Ramsey Theory (on the whole), and on the theory of arithmetic structure in the integers. I would characterize my interest in mathematics, in general, as very broad. I enjoy the interdisciplinary nature of certain types of mathematics, and appreciate collaboration between mathematicians with diverse interests. I also take special interest in developing my skills as an educator and as a member of the academic community and workplace. I was the graduate coordinator for the DIMACS REU 2010-2013, and I participate in many other projects and endeavors to develop my skills as a teacher, a mentor, and an administrator.

2 thoughts on “Research in Mathematics”

  1. A math Ph.D. student I knew in college claimed he only had about five hours per day in which he could really work — the rest of the time was important for ideas to percolate and develop subconsciously in his brain. I’ve observed a similar pattern in myself, especially when I’m trying to solve a difficult problem: I can only consciously force myself to think about it for so long in one sitting, but often the real insights come at other random times (e.g., trying to go to sleep). What are your thoughts on this?

    By the way, as a physicist I am obligated to point out that the picture captioned “Mathematics” is really physics (albeit expressed as mathematics)!

  2. I agree entirely with the notion that, if doing some kind of intensive research work, 8 hours per day on that work alone will probably be less efficient than 5 hours per day or something (plus 3 hours of whatever else — there are plenty of other things to do). I can sit down and work for eight hours on computer code, grading, reading general-interest articles, preparing for teaching, etc. But when solving a problem, it’s important to set aside time (the right amount of time) to do that intense sort of work — and it is often is successful. But breaking it up with other work (the type above), or with discussions with a colleague (collaborator or just a “sounding board”).

    And sometimes the spark of insight happens on the drive home or in the shower. I’ve heard (and experienced) anecdotes where that epiphany happens while doing something like driving, showering, watching TV, jogging, etc. I believe this is called the “shower effect” (or at least, I read about it once and a web search produces at least these two pages: The Shower Effect and The Creative Process). I think these moments of clarity wouldn’t happen without all the intense work time, so I have not yet changed my research methodology to a routine of driving up and down 287 and showering every 5 hours… although I might consider it, maybe for a week, just to see if it works.

    And as a mathematician, I am obligated (as stubbornly and unnecessarily as possible) to assert that differential equations and Fourier transforms are mathematics, and that using them to predict or model structures in physics would be an exercise in applied mathematics. Such modeling not being pictured, I will amicably stand by my caption ; )

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