## 05. Doing mathematical research

### Practical advice

Terry Tao, a Fields’ Medalist, has advice for students starting out in mathematical research:

1. Solving mathematical problems

2. There’s more to mathematics than grades and exams and methods

3. Ask yourself dumb questions – and answer them!

4. Write down what you’ve done

6. Work hard

7. On writing

8. Write a rapid prototype first

9. Continually aim just beyond your current range

- Terry has written:
*“… it is necessary to “cut one’s teeth” on the non-glamorous parts of a field before one really has any chance at all to tackle the famous problems in the area; take a look at the early publications of any of today’s great mathematicians to see what I mean by this.”*

### Reflections on mathematical research

“*Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house.*” Henri Poincare, La Science et l’hypothese.

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“*… in the old days, to get numerical results you had to make enormously drastic simplifications if your computations were done by hand, or by simple computing machines. And the talent of what drastic simplifications to make was a special talent that did not appeal to most mathematicians. Today you are in an entirely different situation. You don’t have to put the problem on a Procrustean bed and mutilate it before you attack it numerically. And I think that has attracted a much larger group of people to numerical problems of applications …*” Peter Lax, Abel Laureate.

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Question for Sir Michael Atiyah: “*How do you select a problem to study*?”

“*I think that presupposes an answer. I don’t think that’s the way I work at all. Some people may sit back and say, ‘I want to solve this problem’ and they sit down and say, ‘How do I solve this problem?’ I don’t. I just move around in the mathematical waters, thinking about things, being curious, interested, talking to people, stirring up ideas; things emerge and I follow them up. Or I see something which connects up with something else I know about, and I try to put them together and things develop. I have practically never started off with any idea of what I’m going to be doing or where it’s going to go. I’m interested in mathematics; I talk, I learn, I discuss and then interesting questions simply emerge. I have never started off with a particular goal, except the goal of understanding mathematics.*” (Michael Atiyah)

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*“I think of a theorem – that this must be true – and then try to prove it. You need to see the whole way through: that is how I tend to work. I try to sketch a proof, and complete a little bit and go home happy. That’s how I work: I try to see the whole thing and then come back and fill in the details. You need the whole community of mathematicians: we also teach, and mathematics is a conversation that goes backwards and forwards between people, and also between the generations. I think that’s important — to have this continuity of ideas.” *Ulrike Tillmann, Oxford University

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” *A series of books aimed at making transparent the thought processes that led to certain results could show to young mathematicians why research is not as impossible as it looks and convey much more efficiently the tricks used by those with much more experience*.” Timothy Gowers, Rough structure and classification. In *Geometric and Functional Analysis*. Birkhauser , 2000.

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“*Mathematicians tend to divide into two groups: there are people who think algebraically and think in equations, and then the other half who think in pictures, and I’m one of them. If I’m thinking about a problem, then I need some sort of 2-dimensional picture to get me going, even if the actual problem is in some large number of dimensions, an arbitrary number of dimensions, or an infinite number of dimensions. If I’m just trying to get going on a problem, then what I need to do is to draw some diagrams, to draw some pictures*.” Frances Kirwan, Mathematical Institute, Oxford.

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“*I really do enjoy doing things that are useful, and where I think mathematics has a real role to play. But it is also a question of learning and understanding. That’s what really drives me – learning new things.*” Valerie Isham, University College, London

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“*I ‘m good at sticking at things. I don’t see things as quickly as some people, but I’m fairly determined, and I’ll get it out somehow. I probably do have a feel for which examples will work and trying them. For this stuff I’ve been doing, the papers have been very long, and it has been a question of really slogging it out and being quite careful with the estimates and things. That suits me: that’s the sort of person I am, stubborn. To write it all down and prove everything rigorously you have to have pages and pages of arguments, but you can usually get a rough estimate out much more easily. I would be working on that much broader scale and getting a feeling for it and being pretty sure that this is going to work, and then comes that awful moment, when you realize that you have to write it all down carefully. I’m aware of being in an area where not that much has been done before, so you have a reasonable number of fairly interesting questions left to have a look at. It was quite satisfying having this dimension thing open and no-one having thought about it. It was really nice to feel that there were big questions there, just waiting to be looked at, really.*” Gwyneth Stallard, Open University

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“*Mathematicians traditionally love to solve problems by thinking. Myself, I hate to think. I love to meta-think, try to do things, whenever possible, by brute force, and of course, let the computer do the hard work. …Traditionally there was a dichotomy between the context of discovery, that nowadays is mostly done by computers, and the context of verication that is still mostly carried out by humans. In my style of experimental math, the computer does everything, the guessing and the (rigorous!)proving, if possible completely seamlessly without any human intervention. Feel free to browse my website for many examples.*” (Doron Zeilberger, Rutgers – what_is_experimental_mathematics_zeilberger)

### Philosophy of research

The following words of Ed Witten should be borne in mind by all students and faculty working on computational mathematics research projects.

In answer to the question (CNN June 27, 2005) of what he does all day, Ed Witten replied:

*“There isn’t a clear task. If you are a researcher you are trying to figure out what the question is as well as what the answer is.*

*You want to find the question that is sufficiently easy that you might be able to answer it, and sufficiently hard that the answer is interesting. You spend a lot of time thinking and you spend a lot of time floundering around.”*

Ed Witten is one of the world’s most highly cited academics, a leading theoretical physicist, but also a Fields’ Medalist, so he knows a thing or two about mathematical research.

Most students, and faculty, have a preference for a certain type of mathematics, a certain way of working, and a certain attitude to problems and their solution. For example, some people prefer algebraic methods and results over analytical methods. Others prefer combinatorial arguments and problems, and others love number theory. Some people like very applied problems, and others like problems that are more associated with mathematical structure, such as problems in graph theory, algebraic geometry, or number theory.

Some people prefer plunging into a calculation while others like to contemplate a proof through intuition and insight.

However in recent years it has become increasingly clear that there is no single right way to do mathematics, and no single formula for successful mathematical research. A productive attitude to research is to seek connections everywhere.

Jurgen Moser, commenting on approaches to mathematical research, wrote:

*“… it seems idle to argue whether to prefer solving of challenging problems, building abstract structures, or working on applications. Rather, we should keep an open mind when we approach new problems, and not forget the unity of mathematics. “*

Computational mathematics is not simply number crunching. It is a computational approach to insight, and that insight leads naturally to proof. In relation to proof and insight Alexandre Grothendieck wrote:

*“What my experience of mathematical work has taught me again and again, is that the proof always springs from the insight, and not the other way round – and that the insight itself has its source, first and foremost, in a delicate and obstinate feeling of the relevant entities and concepts and their mutual relations. The guiding thread is the inner coherence of the image which gradually emerges from the mist, as well as its consonance with what is known or foreshadowed from other sources – and it guides all the more surely as the “exigence” of coherence is stronger and more delicate.”*