Understanding mathematics

John von Veumann:

Young man, in mathematics you don’t understand things, you just get used to them.

My favourite mathematical quote of all time. It exactly captures the flavour of so many things, not just in mathematics, but also in life.

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Classifying the classics

There is a very small number of scientific papers that are agreed-upon classics.

I won’t stir much controversy to suggest as examples Shannon’s papers on information theory, Turing’s paper on computation, and Einstein’s papers on relativity.

Further down the totem pole, every subfield of science has its own classics. One subfield I work in – quantum information science – has Feynman ’82, Deutsch ’85, Bennett et al ’93, Shor ’94, and so on.

Among the crowd of people who work in the field, not only are journal references redundant in that list, the list itself is almost redundant. Everybody in the field already knows what the classics are, and would probably write down much the same list, albeit in a rather more complete fashion.

I’m curious as to what makes a paper a classic, and if there are broad classes of classic papers.

One thing I find really striking is that many classic papers do not smash really difficult problems. In physics – both theoretical and experimental – the stereotype of a major advance is the solution to some long-standing problem. Yet only a few of those classic papers solve a long-standing problem.

So what do those other papers have that give them classic status?

I’ll come back to this question in more detail at some later time. For now, a short answer.

In at least some instances, classic status is accorded a paper that identifies a hitherto unknown motivating context – a big story you can tell about why some set of big questions is interesting. The paper might ask some interesting new “big” questions itself, or set some old questions in a new context that makes it apparent why those questions are interesting. Furthermore, the paper will suggest a framework for making progress on those questions, often by introducing new definitions, and making some minor technical advances.

For example, in 1985 Deutsch introduced his model of a quantum computer, and showed how to solve a simple problem in that model. Technically, this was not difficult. But it showed people a way to make further technical progress, progress that resulted in Shor’s famous 1994 paper on fast factoring with a quantum computer.

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A simple question

Here’s a question that generated a lot of interest among the people involved in my metals and superconductors short course.

The question is this: Is Newton’s second law – that the net force on a body is equal to its mass times its acceleration – simply a mathematical definition of what a force is? Or is there some additional physical content? If so, what precisely is that content?

Some people regard the answer to this question as “obvious”. As is often the case with interesting questions, one person’s “obvious” may be another person’s “wrong”, so I’d be interested to hear other people’s opinions. If I have time (very busy the past couple of weeks) I’ll put together my own best understanding of the answer.

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Lectures on Metals and Superconductors

PDF lecture notes for an informal seminar series on metals and superconductors. Files are typically 10-20 pages long, and between 1 and 2 megabytes, with a few somewhat larger files.

Part I: The Drude Theory of metals

Lecture 1: Introduction to the Drude theory of metals
Lecture 2: Applications of the Drude theory
Lecture 3: AC electrical conductivity of a metal
Lecture 4: Thermal conductivity of a metal

Part II: The Sommerfeld theory of metals

Lecture 5: Ground state properties of a free electron gas
Lecture 6: The Fermi-Dirac distribution
Lecture 7: Energy density and specific heat in a free electron gas
Lecture 8: The Sommerfeld model of metals

Part III: Superconductivity

Lecture 9: Superconductivity: basic facts

Lectures on Metals and Superconductors

PDF lecture notes for an informal seminar series on metals and superconductors. Files are typically 10-20 pages long, and between 1 and 2 megabytes, with a few somewhat larger files.

Part I: The Drude Theory of metals

Lecture 1: Introduction to the Drude theory of metals
Lecture 2: Applications of the Drude theory
Lecture 3: AC electrical conductivity of a metal
Lecture 4: Thermal conductivity of a metal

Part II: The Sommerfeld theory of metals

Lecture 5: Ground state properties of a free electron gas
Lecture 6: The Fermi-Dirac distribution
Lecture 7: Energy density and specific heat in a free electron gas

Recruiting PhD students

How ethical/sensible is it to take on large numbers of PhD students?

The question has been on my mind, as my last week was spent helping organize a “Postgraduate information and recruitment day”, with the event itself held last Friday.

We brought eight potential PhD students from outside UQ into the Department of Physics, and gave them lots of chances to mix with the various research groups.

It was a fun day for all, and hopefully we’ll end up with some great new PhD students as a result.

It also returned to the forefront of my mind some ethical questions about taking on PhD students.

Traditionally, PhD degrees in many of the Arts and Sciences have focused almost entirely on preparation for a career in academic research or teaching.

(Incidentally, much of what I say below does not apply to professional postgraduate degrees, such as are found in many Engineering and Business Departments, which often have a substantial focus on the so-called real world.)

The stereotypical “ultimate success” story in such degrees is of the PhD student who writes lots of papers and becomes a bigshot Professor, with many students of their own.

Conversely, there is a slight stigma associated with people who don�t get postdoctoral positions (or, similarly, postdocs who don�t get tenure-track positions, and so on up the chain), who leave to go into industry, or who leave their field altogether. A not unusual presumption is that a person who does this has �failed�. People speak of it with regret, with pity, or, surprisingly often, with disapproval or even scorn.

Of course, the bigshot Professors typically have many PhD students, sometimes as many as dozens over a career. As a result, either the number of academic positions needs to expand at an incredible rate � which it did for several decades after World War II (perhaps the reason for our current culture), but is not doing any longer – or a heck of a lot of students are going to �fail� according to now-current criteria.

This, in my opinion, is an appalling situation. What can be done about it?

As I see it, there are two broad options. The first option is to change the culture so that �success� for a PhD student is redefined in an expanded way. The second option is to greatly reduce the number of PhD students.

I�ll talk briefly about how each of these options might be taken, before making a comparison. The pictures I paint are perhaps somewhat unrealistic, partially because of the brevity, and partially because of my incomplete knowledge. Nonetheless, I think the pictures I sketch here might serve as a useful basis for improving the PhD experience.

How might we expand the definition of success for PhD students?

Imagine that each semester PhD students are given the option of working, one day per week, as consultants on a variety of real-world projects, for real companies.

A student who spends perhaps two or three semesters so engaged might work on five or six such projects, and might make 20-30 contacts at a dozen or more companies.

Such students would get a good sense for how their skills may be applied in the real world, and for much they�d enjoy doing such work, as compared with research. They�d gain confidence that they can succeed in such an environment. Perhaps most important, they�d gain a sense that they can go out and get themselves jobs � and they�d have a starting list of contacts to help them get jobs.

I believe such a consulting scheme would have the further benefit of greatly reducing the stigma associated with not going on with a career in basic research. The people doing postdocs and searching for faculty posts would be doing so fully informed of the alternatives available to them. In short they�d be making a conscious choice to take the academic route. This would both reduce the number of people searching for academic positions, and improve the quality of life for all involved.

A variety of objections may be raised to such a consulting scheme.

Could such a consulting scheme be operated across all disciplines?

In some disciplines, such a consulting scheme could almost certainly be operated. I�ve never consulted, but am reliably informed that a lot of consulting work is available for physicists, if one knows who to talk too.

What about other disciplines? Certainly, in many disciplines it is easy to see such schemes being practical. Mathematics, chemistry, biology and economics, for example, should all offer ample opportunities.

Other disciplines might be more difficult. It is not so obvious what consulting opportunities would be available to PhD students studying mediaeval history, or the romance languages. Nonetheless, such students often have fine writing, communication and analytic skills � skills that are in high demand. It is possible � though I don�t know � that consulting opportunities could be organized for such students. If not, then I have few suggestions to make, other than to reduce the number of PhD students in such programs. This option is described in more detail below.

Won�t such a consulting scheme distract the students?

Most Universities currently allow their students to work 1-2 days per week as tutors or teaching assistants. If they gave them the option of working as tutors / teaching assistants or doing consulting, but not both, this ought to alleviate any difficulties with distraction.

Won�t this leave too few tutors or teaching assistants to teach into Departmental courses?

This might be a problem, but could easily be alleviated, perhaps by restricting the pool of students allowed to engage in the consulting work � perhaps only second year PhD students would be allowed to do it.

Won�t it be much more financially attractive for students to consult than to tutor?

It might. On the other hand, a portion of the consulting fee might be used to increase rates for tutors, in order to achieve pay parity.

Who would organize the consulting?

If the Department takes a small percentage of the total consulting fees paid, then they ought to be able to pay somebody, at least part time, to co-ordinate the consulting program.

Might a Department end up exploiting their PhD students to make lots of money?

This would be something to guard against. You could imagine a horror situation where the primary role of grad students in some exploitative Departments is to engage in consulting jobs to fund faculty research. A way of preventing this would be to put severe restrictions on how income raised from consulting could be spent. It might be restricted to spending, for example, on improving student conditions, such as offices, travel, and access to computer equipment.

The other option for improving the PhD experience identified earlier is to reduce the number of PhD students. How might this be achieved?

This is a more difficult option to act on directly, because it is bound up with high-level institutional issues. (Unless, of course, one has a lot of influence at the highest levels! I�m taking the perspective of an individual faculty member here.)

Many Universities, certainly in Australia, but also around the world, provide significant incentives � both carrot and stick � to increase the number of PhD students. For example, Department, School, Faculty and University-wide funding is often directly linked to the number of PhD students. This leads to considerable pressure on individual academics to take large numbers of PhD students.

Combined with these institutional incentives, of course, is the natural desire that many academics have to create large and vibrant groups of their own.

For these reasons, my inclination is to work the other end, concentrating on changing the culture so that PhD students are given the opportunity to voluntarily engage in a small amount of non-academic work.

This leaves a final challenge, which is what to do in those fields where there is little demand for the services of PhD candidates, outside of an academic context. In that case I think the difficulties are far more formidable, and will require difficult long-term transformations, in both faculty and institutional attitudes and funding models, so that large research groups are not seen as an unalloyed good, but rather where the decision to take PhD students is made extremely carefully and cautiously.

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Why Universities take graduate students

John Wheeler probably wins the prize for pithiest summation (and one of the best): “The reason Universities have students is so they can teach the Professors” (quoting from memory)

Via Lance Fortnow, a thought-provoking speech by UCLA Professor Andrew Kahng on the role grad students can play in a University.

Of course, neither of these tells the whole story, by a long shot. But they’re both pretty worthwhile ideals.

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Professional skills for scientists

In an earlier essay I talked about the development of professional skills by research scientists.

Now, this seems like such a self-evidently good thing, what more could there be to say?

Oddly, however, the development of such professional skills is, sometimes, seen in rather a bad odour by some scientists, as though a focus on anything not purely technical is to be derided. I have occasionally heard a scientist criticized or sneered at, just slightly, for giving a particularly polished talk, writing an especially clear paper(!!), or going to the trouble of developing and maintaining a strong professional network. (More often, I hasten to add, people celebrate such achievements.)

I have even – and this is the problem that concerns me – somewhat more frequently heard older scientists counsel younger scientists that they shouldn’t make their presentations look too professional, for just this reason.

The imputation undelying all this seems to be that people who do develop these sorts of professional skills have nothing to do with their time, and that rather than doing so-called “real science”, are merely wasting their time on cosmetic foolishness.

This line of thought is rubbish. It is true that there are a few scientists – a very few – who believe (in deed, if not in word) that form is more important than substance, or that political power plays are more important than asking and answering interesting questions.

However, the majority of people with top-notch professional skills develop them precisely because they realize how important such skills are to effective research.

How much more effective will your research be if you can communicate your ideas in such a clear and compelling fashion that someone from another field can understand your main problems – and maybe contribute new lines of thought that open up new avenues to solution, or suggest problems in their own field that you may be able to contribute to the solution of?

Albert Einstein, often revered as an icon of individualism, and for his focus on pure research, was actually a consummate research networker, writing many thousands of letters (striking for the clarity and care of their prose) to colleagues, describing his latest ideas, and receiving in turn many thousands of letters keeping him up with the latest thoughts of the other leading scientists of the day.

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Why are research skills so seldom taught? (V3.0)

Revised version of earlier entries, in response to comments by Dave Bacon and Ben Toner.

Undergraduate education in physics is usually concentrated on learning certain basic facts about physics, and technical skills that enable one to solve problems in physics. While both these are essential facets of doing research, many other equally essential skills are neglected, or ignored completely.

Why is this the case? In part it may be because not all people taking physics degrees necessarily hope to do research one day. However, to an extent far greater than in almost any other subject, an undergraduate degree in physics is, at least nominally, focused on the task of preparing people for research.

At the PhD level, while there is a strong focus on actually doing research, relatively few supervisors engage in much active discussion of how research is done. If a student is lucky they may see a particular research style modeled, through interactions with their supervisor and other more senior scientists.

Such modeling is potentially quite valuable, especially if a student is exposed to a wide range of research styles. However, what works for one person may not work for others. This is especially true when one person is inexperienced and lacks confidence, while another is very experienced and has considerable confidence. Furthermore, each individual needs to develop their own style, suited to their own combination of talents.

Many students fail even to see such modeling. A remarkably common attitude is that students either “have it”, or “don’t”, when it comes to research skills, and that this justifies neglect of students who “don’t have it”.

This sells students lumped into either category short.

It is true that some beginning PhD students are exceptionally well equipped to do the tasks required of a PhD student. Such students may complete their PhD much more rapidly than usual, with apparently astounding success. However, such students may also plateau – they may never move beyond this level, stagnating instead of growing into a new set of skills beyond that required of a PhD student.

Similarly, other beginning students may be very well equipped in some ways, but lacking in certain essential skills that result in them being placed into the “don’t” category. Might such students benefit from learning some basic research skills?

I believe classes aimed at improving student’s research skills – what we might call “research literacy” classes – can be effectively integrated into both the undergraduate and postgraduate curricula.

Before describing how this integration might be achieved, one comment on what I mean by research skills.

Research skills may usefully be divided into two classes.

The first class is professional skills, such as public speaking, writing technical prose, and networking, which can be learnt and applied outside the context of research.

The second class is technical skills, such as finding and solving good research problems, and determining what constitutes a research result. These are skills that can only be learnt by someone actively engaged in the practice of doing research. The reason is that as yet there isn’t any good general theory of how to do research. Different things work for different people, and there is no test you can take to find out how you should operate. Instead, you need to try different things out, see how they go, and improve from there.

This distinction between professional and technical skills has important implications for the integration of research literacy classes into the undergraduate and postgraduate curricula.

At the postgraduate level, research is usually the primary activity, and research literacy classes could easily be integrated in parallel with actual research. I am currently trying this out on a small scale by forming a discussion group in which students and faculty members discuss the difficulties involved in doing research, and potential solutions to those difficulties. These solutions can then be tried out by members of the group, evaluated individually, and improved upon with the assistance of the entire group.

At the postgraduate level, no distinction need be made between professional and technical skills. However, at the undergraduate level the situation is more complex. At present most undergraduates do not actively engage in research. Instead, most undergraduate programs focus primarily on learning the basic knowledge and problem-solving skills that are seen as necessary, but not sufficient, preconditions to being able to do research.

Given this constraint, research literacy classes for undergraduates would need to focus on professional skills, not technical skills. To some extent this already occurs in some University systems, most notably the US system, with their focus on obtaining a well-rounded liberal arts education. In such a system students in the sciences are less likely to forget how to write an essay or make a public presentation than in a more narrowly focused system such as Australia’s. However, in all systems it seems that considerably more attention could fruitfully be paid to the development of professional skills in undergraduates.

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