Kevin Houston

One for researchers. Last year I, with my colleagues Roger Bielawski and Martin Speight, edited a book, Variational Problems in Differential Geometry. There is currently a US promotional offer for this book:

Willmore surface sculpture

For a surface we can define the notion of curvature in a number of ways. For Willmore surfaces we use mean curvature so let’s define that. At a point of a surface take a normal vector. Any non-zero vector in the line perpendicular to the tangent plane to the surface will do. Next take any plane that contains this normal. The intersection of this plane with the surface is a curve. Take the curvature of this curve at the point , call this curvature . (The curvature of a curve at a point is where is the radius of the circle that best approximates the curve at that point.) Next take the plane containing the normal that is at right angles to the first one. We get another curve and hence another curvature, call it . Take the mean of these two curvatures, . This is the mean curvature at ; we will denote it by . We can do this at every point and so we get a function given by . An alternative viewpoint is given on the Wikipedia page.

Now we integrate the square of this function over the whole of the surface. The Willmore energy of a surface , denoted , is the number . That is we integrate over the whole surface the square of the mean curvature. So for every surface we have a number.

If I deform a surface slightly, then the Willmore energy is likely to change, maybe up, maybe down. A surface is a Willmore surface if any small deformation of the surface cause the Willmore energy to increase. I.e., out of all nearby surfaces this surface has minimal Willmore energy. If we take surfaces with the topology of a sphere, for example a perfectly round sphere, an ellipsoid or an egg, then . The minimum occurs for the perfectly round sphere.

What if we move from spheres to tori? The Willmore Conjecture is that for surfaces with the topology of the torus we have . This conjecture has resisted proof for many years and the recently announced proof looks good to the experts (sometimes people make dubious claims for proofs of famous problems, see NP versus P and the Riemman Hypothesis for examples of this). The proof is rather technical and can be found on the arXiv: Willmore Conjecture paper

For pure mathematicians the main interest of Willmore surfaces is that you get rather good looking surfaces. However, Willmore energy is of use in applied mathematics as it can be used to describe bending energy, for example in biological problems on the elasticity of membranes.

The sculpture is inspired by a four-lobed Willmore torus and as such is not a Willmore surface, one can see this clearly due to the flat looking parts of the sculpture. A picture of a four-lobed torus can be found at GeometrieWerkstatt. The plaque near the sculpture makes it clear that the shape is only inspired by a Willmore surface as can bee seen below.

Plaque

A video was made of the public lecture by Franz Pedit. I’ll link to that when it appears on the web.

I’ve mentioned elsewhere that the current system of publishing research is flawed. The government, i.e. taxpayers, pays us to do research, we send the resulting papers to journal publishers, we referee the papers and edit the journals for free and then the publishers sell the research back to us for a high price. The result is that the taxpayers pay twice, we work for free and the commercial publishers get rich.

My answer to this was to start charging for my refereeing services. A small change but so far no-one has asked me to referee for something from a commercial publisher so it is really no change! However, a bigger and more effective method of change is coming – an academic boycott against Elsevier, considered by many to be a serious offender in this problem, has been started by Tim Gowers. Read the story over at the Chronicle of Higher Education, home to Prof Hacker.

Within the card magic community the Gilbreath Principle is a well-known but much misunderstood mathematical principle. Few magicians know much about its creator, Norman Gilbreath, and in particular they are unaware of his other mathematical work. Following a recent email conversation with him about the principle (always go to the source!) he kindly sent me an offprint of a recent paper on the Gilbreath Conjecture.

The Gilbreath Conjecture is a conjecture about primes and is fairly easy to state. Consider the sequence of primes
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …
Now work out the (absolute) difference between neighbouring terms
1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, …
Work out the absolute difference between terms for this sequence and keep doing this:
1, 0, 2, 2, 2, 2, 2, 2, 4, 4, …
1, 2, 0, 0, 0, 0, 0, 2, 0, …
1, 2, 0, 0, 0, 0, 2, 2, …
1, 2, 0, 0, 0, 2, 0, …
1, 2, 0, 0, 2, 2, …
1, 2, 0, 2, 0, …
1, 2, 2, 2, …
1, 0, 0, …
1, 0, …
1, …

The conjecture is that the first term on a line, after the first line, is always a 1.

Gilbreath’s paper, Processing process: The Gilbreath conjecture, has recently been published in the Journal of Number Theory. (You can find it at http://dx.doi.org/10.1016/j.jnt.2011.06.008 but unless you have a subscription to the journal you will have to pay for it.)

The introduction states the following

There is one very important aspect of history that is often left out – the process. This is even true of the history of mathematics. I will give an example. A number of years ago (around 1958) I developed a number theory conjecture concerning the primes. This is known in number theory as the Gilbreath conjecture. It is easy to state but even though the great number theorist Erdos believed it was true, he also believed it would take about 200 years to prove.

Who am I to doubt Erdos? However, it seems no one has seriously considered why and how I
came up with this conjecture. So, I will now describe the process, and some observations suggested by this process – to hopefully show why processing process is important.

If you are interested in the conjecture, then see Wikipedia, Facebook, Wolfram demonstrations, and a request for help by a poster on the xkcd forum.

Variational Problems in Differential Geometry cover

Along with my colleagues Roger Bielawski and Martin Speight, I have edited the proceedings of the Workshop on Variational Problems in Differential Geometry which we held here at Leeds in 2009. The book has just been published by Cambridge University Press in the prestigious London Mathematical Society Lecture Notes Series.

It’s intended for researchers working in differential geometry and can be bought from the publishers or via Amazon.co.uk or Amazon.com.

Needless to say, it’s a must-have if you work in the area. And if you work in a university, then don’t forget to recommend it to your library! Thanks.