Lyapunov-Schmidt reduction

At an early age we learn how to tackle the problem of solving linear equations for unknowns. What about solving nonlinear equations for unknowns? In general there is not much which can be said. One promising strategy is to start with a problem whose solution is known and perturb it. Consider an equation where should be thought of as the unknown and as a parameter. The mapping defined by is assumed smooth. Suppose that so that we have a solution for . It is helpful to consider the derivative of with respect to at the origin. If the linear map is invertible then we are in the situation of the implicit function theorem. The theorem says that there exists a smooth mapping from a neighbourhood of the zero in to such that . It is also (locally) unique. In other words the system of equations has a unique solution for any parameter value near zero.

What happens if is degenerate? This is where Lyapunov-Schmidt reduction comes in. Suppose for definiteness that the rank of is . Thus the kernel of is one-dimensional. We can do linear transformations independently in the copies of in the domain and range so as to simplify things. Let be the standard basis in a particular coordinate system. It can be arranged that is the span of and the range of is the span of . Now define a mapping from to by where is the projection onto the range of along the space spanned by . Things have now been set up so that the implicit function theorem can be applied to . It follows that there is a smooth mapping such that . In other words satisfy of the equations. It only remains to solve one equation which is given by . The advantage of this is that the dimensionality of the problem to be solved has been reduced drastically. The disadvantage is that the mapping is not known – we only know that it exists. At first sight it may be asked how this could possibly be useful. One way of going further is to use the fact that information about derivatives of at the origin can be used to give corresponding information on derivatives of at the origin. Under some circumstances this may be enough to show that the problem is equivalent to a simpler problem after a suitable diffeomorphism, giving qualitative information on the solution set. The last type of conclusion belongs to the field known as singularity theory.

My main source of information for the above account was the first chapter of the book ‘Singularities and groups in bifurcation theory’ by M. Golubitsky and D. Schaeffer. I did reformulate things to correspond to my own ideas of simplicity. In that book there is also a lot of more advanced material on Lyapunov-Schmidt reduction. In particular the space may be replaced by an infinite-dimensional Banach space in some applications. An example of this discussed in Chapter 8 of the book. This is the Hopf bifurcation which describes a way in which periodic solutions of a system of ODE can arise from a stationary solution as a parameter is varied. This is then applied in the Case study 2 immediately following that chapter to study the space-clamped Hodgkin-Huxley system mentioned in a previous post.

Sydney Brenner on mice and men

On 27th September 2007 I had the privilege of hearing a talk by Sydney Brenner in Berlin. Brenner is a Nobel prize winner (2002) and known for his role in deciphering the genetic code, discovering messenger RNA and launching the humble worm Caenorhabditis elegans on its brilliant career as a model organism. I find that Brenner is an inspiring speaker and I had the impression of experiencing a very special source of knowledge and an exceptional individual. The talk was seasoned with critical comments on various aspects of modern molecular biology including a tough one-liner directed at systems biology. He also had something more general to say about the applications of mathematics to biology – unfortunately I do not remember the details. In any case, his variety of scientific argumentation reminded me of some of the best things about the way mathematics works.

Yesterday I discovered online videos of two talks of Brenner. I watched them and was glad I did. The first begins by talking about the redshift as a tool which can be used to obtain information about the distant past of our universe and asking if there exists something similar which could be used to explore the distant evolutionary past of life. Brenner points out that the genomes of many organisms have now been deciphered and asks what kind of information can be obtained from them. He mentions the following ‘inverse problem’. Suppose we were given just the genome of an organism without knowing the organism itself. Could we then reconstruct that organism? It seems that this is far beyond what can be done at the present time. He then goes on to discuss the question of comparing the genome of different organisms and trying to define a kind of evolutionary distance between them. As a concrete example he takes the case of mice and men.

One of the main themes of the lectures is finding ways of determining the speed at which the genome is evolving in different species. He points out that the genetic data contain no arrow of time. Thus they are equally consistent with fish evolving into human beings or fish having been formed by degeneration of previously existing human beings. The external facts that allow us to decide in which direction evolution goes come from the fossil record. The data show that the mouse genome is evolving much faster than the human genome. Getting this kind of information requires comparing the genomes of more than two species.

In what way is it possible to obtain information about the evolution of the genome? Certain types of statistical analysis of the occurrence of different bases in the genomes of different species can do this. Brenner emphasizes the important of silent mutations, those where a change in one base replaces a codon by another one corresponding to the same amino acid. An advantage of studying these mutations is the absence of selective pressure on them. What this statistics involves is anything but applying standard (perhaps powerful) methods of analysis. It rather has to do with having good ideas about what patterns to look for in the data. Brenner points out that since the genome data are freely available on the internet and the computing power required is modest it would be possible to develop home genomics. That is to say: someone could develop important new ideas for the analysis of the dynamics of the genome by playing about on their home computer.

The Fraunhofer Society

In Germany there are several large research organizations, among them the Max Planck Society (which happens to be my employer) and the Fraunhofer Society. Yesterday I went to the inaugural lecture of Ulrich Buller at the University of Potsdam. He is the head of research within the Fraunhofer Society and the theme of his talk was the research strategy of the society. I thought it would be a good opportunity to improve my knowledge of local (and national) scientific politics.

The annual research budgets of Fraunhofer and Max Planck are 1.2 and 1.3 billion euros respectively. The Fraunhofer society has 54 research institutes spread over Germany and over different research fields. The Max Planck Society has 80. (The counting here may not be consistent since it depends on distinguishing between individual institutes and branches of other institutes.) The mission of Max Planck is fundamental research while that of Fraunhofer is applied research. Another big difference between the two organizations is that while the great majority of the funding for Max Planck comes from the government a Fraunhofer Institute has to earn a large part of its money from contracts with industry, after a short honeymoon period. Buller emphasized that a natural consequence of the need to earn money is that a Fraunhofer Institute has to be run in a way which has similarities with the way a private company is run. For this reason, he said, the director of a Fraunhofer Institute must be given a large degree of autonomy in deciding on the direction of research done by his staff. It also means that there is inherent competition between the individual Fraunhofer Institutes which should not be suppressed. At the same time it makes sense to have cooperation between the institutes and this leads to a certain conflict of goals. To try and address this issue the institutes are arranged into groups (Verbünde) with related research fields which try to foster cooperation where it is useful.

One of the most lucrative developments arising from research within the Fraunhofer Society concerns the technology for the MP3 player which was developed at the Fraunhofer Institute for Integrated Circuits in Erlangen. It was clear from the talk that the license fees connected with this are an important and welcome resource for the Society.

A point which was stressed by the speaker was the relatively small percentage of Germans who are qualified to take up jobs in scientific research. (He quoted a figure of 12%.) Here ‘relatively’ means ‘relatively to a number of other countries’. A consequence is that, at least within the national market, Max Planck, Fraunhofer, the universities and other research organizations are competing for a relatively small pool of potential employees. He talked about some of the ways that the Fraunhofer Society is trying to improve the situation from its point of view. He mentioned initiatives to set up collaborations between the different players (Fraunhofer with universities, Fraunhofer with Max Planck etc.) and thus better use the available human resources.

Where is mathematics in all this? It is plausible that there is money to be earned with computer science, but with mathematics? Remarkably there is a Fraunhofer Institute which does mathematics (Fraunhofer Institute for Industrial Mathematics) and it has been extremely successful. The original scepticism of the Society was overcome by a generous contribution from the state of Rheinland-Pfalz (Rhineland-Palatinate). In fact the institute was so successful financially that reponsibility was taken over by the Society earlier than originally planned. This success story is told in an article by the institute’s founding director Helmut Neunzert in Mitteilungen der deutschen Mathematiker-Vereinigung, 4, 262-268. It is impressive that he launched this project when he was already 59. Of course in the end the products sold by the institute do have a lot to do with the use of computers but the real mathematical content is enough to demarcate it from other Fraunhofer Institutes situated squarely in computer science.

In memoriam Jürgen Ehlers

Today I attended the funeral of Jürgen Ehlers who died recently and quite unexpectedly for those around him. Here I want to collect some personal thoughts about Jürgen. I was associated with him for twenty years during which time he was my colleague and mentor. I was first a member of his research group in Munich and later at the institute in Potsdam (Max Planck Institute for Gravitational Physics) in whose founding he was the key figure. This is all the more remarkable since his passion was for science and not for scientific politics.

This is not the place to write about the personal qualities which I admired in Jürgen and I will confine myself to talking about some of his scientific qualities. Although he decided to go into physics rather than mathematics at a very early stage of his career, he had a keen sense for mathematical questions and a wide mathematical knowledge. He was not the kind to devote himself to the production of long and complex mathematical proofs. What he did was to identify important physical questions and the challenging mathematical problems which lay hidden beneath them. Once he had identified them he was a master of exposition whose explanations were a pleasure to listen to. He came back again and again to statements which are frequently made more or less uncritically in textbooks on general relativity such as ‘Newtonian gravitational theory is the limit of general relativity as the speed of light tends to infinity’ or ‘It follows from the Einstein equations that small bodies move on geodesics’. He saw that these phrases were in need of not only a proof but of a precise statement. He contributed to the first of these questions by formulating his ‘frame theory’ and also worked hard on the second. His efforts are now beginning to bear mathematical fruits in the form of rigorous theorems on these questions.

In doing science Jürgen emphasised making distinctions, e.g. ‘distinguish between a mathematical model in physics and its physical interpretation’, ‘distinguish between an intuitive argument and a rigorous proof’, ‘insist on precision in terminology where it can bring clarification’. These things may seem obvious but they are often enough neglected. In communicating these things to me he was preaching to the converted but he encouraged me by his example to propagate these same ideas. I remember him once talking about the difference between the concept of proof in physics and mathematics. To establish a mathematical fact one proof suffices. Additional, more or less independent, proofs may bring more insight but they are not necessary. The physicist, on the other hand, often likes to have several proofs. He likes to reach the same endpoint by various routes. Each of the two procedures has its own advantages. Very roughly speaking, the physicist’s approach takes one fast and far. The mathematician’s approach has the advantage that the gains it achieves are lasting. What I have said about distinctions might sound like abstract philosophy but my experience is that these ideas are of great worth in improving communication between physics and mathematics. No doubt similar things are true for the relations between other sciences.

Liesegang rings

Liesegang rings are a phenomenon which is widely known due to the work of Raphael Eduard Liesegang in 1896 although he was probably not the first one to make such a observation. An experimental set-up which can be used is the following. A Petri dish is covered with a layer of a gel containing potassium dichromate. Then a drop of silver nitrate solution is deposited at the centre of the dish. It is observed that over a period of hours coloured rings appear which are centred on the point where the drop of solution was. It turns out that similar observations can be made in many other chemical systems. Another typical set-up is to take a test tube filled with gel containing one chemical and put a solution of another chemical on top of the gel. In this case horizontal bands are produced. Part of the fascination of the Liesegang phenomenon is the striking visual patterns which accompany it. Of course it should not be forgotten that not everything which looks the same must have the same underlying mechanism. This kind of phenomenon appears to be widespread in chemistry and has also been invoked in connection with certain biological phenomena. See for instance the discussion and pictures of fungal growth in http://seedsaside.wordpress.com/2008/02/21/liesegang-rings/

It is tempting to try and find a mathematical explanation of the Liesegang phenomenon. It has been observed that the position of the rings follows a geometric progression, that the time of their appearance goes as the square root of the distance and empirical laws for their thickness have also been stated. Thus there are some definite things which a mathematical model could try to reproduce.
There have been many attempts to give a theoretical (and mathematical) account of the effect. Here, for reasons of personal preference, I will concentrate on models which can be formulated in terms of systems of partial differential equations.

On 6th May Arnd Scheel from the University of Minnesota gave a talk on Liesegang patterns at the Free University in Berlin. I had already been interested in the subject for several months and so I naturally wanted to attend. Due to an overlap with the time of my course on general relativity I was only able to hear about half of the lecture but fortunately the speaker took time to explain some of his insights into the subject to me afterwards. The models he uses are reaction diffusion equations, i.e. systems of the form where the unknown is vector-valued, is a diagonal matrix and is some smooth function. Here the spatial dimension is one so that the model is describing bands rather than rings. There are models in the literature where the function is very irregular (in particular discontinuous). Here I want to restrict to the case that is smooth. Rather than trying to describe the dynamics of the formation of the rings it is easier to concentrate on the final steady state. Then the time derivative of vanishes and the equation reduces to a system of ordinary differential equations which can be studied using methods from the theory of dynamical systems. One of the main things I learned from Scheel’s talk was that the existence of Liesgang patterns can be associated to the presence of homoclinic orbits in this dynamical system. In other words, there is a time independent solution of the dynamical system and another solution which converges to it both in the past and in the future. The Liesegang bands correspond not to the homoclinic orbit itself (which would just give a single band) but to other solutions of the dynamical system which approach it asymptotically. Whether appropriate solutions exist depends on the form of and the chemical literature contains a huge variety of choices. Since the phenomenon is so widely observed it must expected that it has a certain stability. If is perturbed a little then the bands should survive. This is somewhat surprising due to the fact that homoclinic orbits are usually not stable under general perturbations of a dynamical system. The explanation proposed by Scheel is that the systems of relevance for chemistry have a special structure which has a definite chemical significance. Only perturbations should be considered which preserve this additional structure and it can happen that perturbations of this special type do not destroy the homoclinic solution.

The special structure of the dynamical systems just mentioned reminds me of a comment made by Karl Sigmund in a plenary talk he gave at the International Congress of Mathematicians in Berlin in 1998 This talk was one of the factors which got me interested in mathematical biology. Sigmund’s point was that the general theory of dynamical systems is not well adapted to many of the problems arising in population dynamics. For good reasons the general theory concentrates on generic systems, i.e. on those whose qualitative properties are preserved under small perturbations. Since experimental measurements are never exact this makes good sense from the point of view of the applicability of the results. It may, however, happen that there is some aspect of the system which should be exempted from perturbations since it has a definite meaning for the applications of interest. In other words, it makes sense to consider systems whose properties are stable under perturbations which are generic modulo an invariant manifold which is kept fixed during the perturbation. In population models whis has to do with the fact that the population of some species being zero has an absolute significance which is independent of the details of the population dynamics. This reminded me of a research interest of my own, cosmology, where the aim is to produce simple models of the dynamics of the universe which also give rise to dynamical systems. In that case if the extension of the universe in one spatial direction tends to zero as the big bang singularity is approached than this has an absolute significance like the extinction of a species in a population model. Restricting to perturbations which preserve some submanifold tends to make homoclinic and heteroclinic solutions more common.

My first contact with Liesegang rings came through a talk by Benoit Perthame I heard at a conference on chemotaxis which took place at the Radon Institute in Linz last December. He mentioned some work on modelling a rare neurological disease called Balo’s concentric sclerosis. In the literature on this subject I found that this disease is characterized by alternating concentric rings of damaged and relatively intact tissue and that a connection to Liesegang rings had been suggested. I intend to return to this topic in a future post.

First steps

Here I am, taking my first steps into the blogosphere. I feel that I am supported by a good title, Hydrobates. This comes from the scientific name Hydrobates pelagicus of the Storm Petrel (as it was called in my youth) or European Storm Petrel (as it is known in these times of globalization). This is a small bird which is about the size of a swallow and spends most of its life on the open ocean. It normally comes on land only at night and only to nest.

There are two reasons why I have chosen this name. The first is that I find the character of this bird inspiring. For me it symbolizes what I regard as an ideal of how a human being can be. It is intrepid. It is confronted with difficult conditions, wind, waves and storms, and turns them to its own advantage. I wish I were more like it. A picture of the bird showing its typical appearance can be found on a Wikipedia web page.

The other reason comes from my experiences as a schoolboy. At that time I got involved in ringing birds. One of the most memorable experiences I had at that time was travelling to small, sometimes uninhabited, islands to ring Storm Petrels. We would put up mist nets during the night to catch them. As a defence mechanism the birds spit a strong-smelling fluid. In handling many of them over the course of a week we found that our clothes became impregnated with the smell of this oil. I came to like the smell but for anyone else confined with one of us in a small space it was overpowering. I still have a vivid memory of the softness of the birds feathers and their curiously shaped foreheads which have something slightly alien about them. The best island for Storm Petrels we visited was Auskerry. On a clear day the lighthouse of Auskerry can be seen from the main island of Orkney where I grew up and before I had been there it was a kind of mythical place for me. In a way it still is now. I have very strong positive associations with the Storm Petrel.

Why have I started to write a blog? Just the other day my first book was published. It is called ‘Partial Differential Equations in General Relativity’. Now that that task has been accomplished I have the urge to communicate in a different way with a wider audience. My enthusiasm has been fired by the blog of Terence Tao. I do not hope to do anything remotely comparable but that example has created in me the wish to do something with this medium.