Prospects in Mathematics


I’ve just spent two days in York, at the annual Prospects in Mathematics meeting.

This LMS-supported meeting is aimed at undergraduates or masters students who are thinking about doing a PhD, and tries to give them information about what research in mathematics is like, what different branches of mathematics entail, where you might go to work on a topic you are interested in, and so forth.

Apart from the mathematics talks, there were presentations about Centres for Doctoral Training, about what life as a PhD student is like (by three local PhD students), and a question and answer session; a reception, and two very good lunches and a dinner (York’s attempt to persuade people to apply there for their PhD?)

As always at these events, I got a very strong impression about the great enthusiasm of the students, as well as their clear-eyed view about what they are letting themselves in for. When I am talking to prospective PhD students, I often find it necessary to warn them that a PhD is not necessarily a passport to a well-paid job. I didn’t mention this once, since the students seemed aware of that; they were there because they loved mathematics and wanted to do more.

A little about some of the more traditional talks.

Tim Spiller talked about quantum information theory. Three important aspects of quantum mechanics are fairly directly connected to potential applications: superposition is what would allow a quantum computer (were one ever built) to solve certain kinds of problem exponentially faster than a classical computer; uncertainty allows detection of unauthorised interception of communications and allows quantum cryptography (in the form of key distribution); and entanglement allows the use of quantum techniques in sensing and imaging. The UK is currently putting substantial resources into this area, and they are hoping to have practical devices available quite soon (maybe not quantum computers, however).

Vicky Henderson talked about the meeting point of economics and psychology. Psychologists know that we are not rational agents, although economists still mostly assume that we are. She talked about prospect theory, proposed by Tversky and Kahneman some time ago, which modifies the utility and probability functions of classical economics which incorporates the fact that we tend to be risk averse on gambles unless the chance of success is very small, in which case we overestimate this chance. The modifications seemed a bit unmotivated to me, and I saw no account of the fact that for some people gambling seems to have positive utility.

Sarah Waters talked about fluid mechanics applied to tissue engineering: a nice talk, but not my thing, I’m afraid.

Katrin Leschke started with soap films, which form minimal surfaces (minimal area for given boundary). She explained that these are harmonic, and so are real parts of holomorphic null curves, with an integral representation in terms of Weierstrass data. She also brought good news about the (threatened) mathematics department at Leicester.

Martin Hairer told us that probabilists are good at deriving results for given probability distributions, but that choosing these distributions is more problematic. The guiding principles are symmetry (e.g. the six outcomes from a well-made cubical die should be equally likely) and universality (the distribution shouldn’t depend on details of the random events causing it). The classical examples of universality are the central limit theorem and Brownian motion; we learned some interesting history of the latter (for example, it was discovered by Ingenhousz half a century before Brown; and Bose, the manufacturer of noise-cancelling headphones, was founded by a student of Norbert Wiener, who gave the mathematical description of Brownian motion as a random function from the Wiener measure). His main interest was a recently discovered universality class described by the KPZ equation, where the most general universality result has not been proved; but he showed us some beautiful simulations, e.g. of dropping Tetris blocks randomly.

Ruth Gregory studies higher-dimensional black holes. While the event horizon of a black hole in ordinary space-time is typically spherical, adding a dimension allows a variety of shapes: spheres, cylinders, tori, and so on. So questions about stability arise. The principles that the entropy of a black hole is proportional to its area, and that entropy cannot decrease, show that ordinary black holes cannot split up into smaller ones; but in five dimensions they can. (So “cosmic censorship” fails in five dimensions.) The cylindrical black holes are unstable, and tend to wobble; it is thought, though not proved yet, that they can break up into spheres with a fractal pattern along the axis of the cylinder. This is the same phenomenon as the flow of water from a tap breaking up into drops as the flow rate changes. I learned from Ruth’s talk that black holes have something in common with Black–Scholes: there are theorems, but these are extrapolated beyond the region where their assumptions hold, by non-mathematicians (physicists or bankers).

Julie Wilson talked about a career which has taken her from a PhD in number theory through crystallography, pattern recognition, and machine learning to metabolomics, food fraud, and archaeology. I learned a new word; “undeamidated”.

Victor Beresnevich talked about Diophantine approximation and metric number theory. Diophantine approximation quantifies and analyses the fact that the rational numbers are dense in the real numebrs, and extends the results to higher dimensions and to manifolds. Metric number theory is mis-named since it is concerned with measures rather than metrics: how big is the set of badly approximable numbers? (Hausdorff dimension 1, but Lebesgue measure 0.) He gave us Khintchine’s theorem, some extensions, and some related open questions.

Finally, my colleague Mark Chaplain told us about the different techniques required in modelling cancer on different scales (within a cell, between cells, or at tissue scale), and recent attempts to construct models which span several scales, aiming eventually at a “virtual tumour”.

The meeting inevitably made me wonder what I would do were I starting a PhD today. I hope I would be brave enough not to go to a CDT; I am not a herd animal. I think what I actually did was what was best for me.


About Peter Cameron

I count all the things that need to be counted.
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