Archive for June, 2010

Black strings in Banff

June 23, 2010

This week I am attending a conference at the Banff International Research Station (BIRS) in the Canadian Rockies. This is an institution modelled to some extent on the Mathematical Research Institute in Oberwolfach which I wrote about in a previous post. Of course there are some differences, e.g. a slightly higher probability of meeting a grizzly bear if you go for a walk in the surroundings. The programme of talks is quite full but in spare moments I have had time to enjoy seeing common North American birds like American Robin and Red-winged Blackbird. Not surpisingly, this area bears no resemblance whatsoever to the town of Banff in Scotland which it is named after.

An interesting mathematical phenomenon is the breakup of a cylindrical configuration into to a row of spherical ones. This plays a role in the study of the collective motion of bacteria. It also comes up in the context of solutions of the vacuum Einstein equations in dimensions greater than four called black strings. A black string is obtained as the product of a spherical black hole (the Schwarzschild solution or its generalization to higher dimensions) with a circle of length L. Today Frans Pretorius gave a talk here about the instability of black strings in five dimensions. Before I write about that I need to present some background.

The Schwarzschild solution is an explicit solution of the vacuum Einstein equations which represents a non-rotating black hole. As for any explicit solution of an equation in physics its physical significance depends on its stability to perturbations. Only configurations which exhibit some stability can be expected to be seen in reality. A complication related to the stability of the Schwarzschild solution is that one type of perturbations which certainly exist are those which make the black hole rotate a little. This results in explicit solution, the Kerr solution. Its existence means that if we want to prove the stability of the Schwarzschild solution we actually need to prove the stability of the Kerr solution. This is a very hard problem. Even the stability of empty space (Minkowski space) in general relativity is hard to prove. A natural strategy to make progress with black hole stability is to look at simpler related problems. One possibility is to linearize the equations around Kerr. This results in a system of linear hyperbolic equations. A further simplification is to replace this system by the linear wave equation. The linear wave equation on Kerr and the boundedness and decay of its solutions has been a focus of attention in mathematical relativity in the past few years. This problem can now be regarded as solved. In a talk yesterday Mihalis Dafermos gave a review of this including, in particular his recent work with Igor Rodnianski. Going from this to the linearized Einstein equations involves new difficulties. Even the formulation of the problem is subtle. Some progress on the problem of the linearized Einstein equations was reported on in a talk of Gustav Holzegel.

It is generally believed that the Schwarzschild solution is stable and linearized calculations supporting this go back a long way. The new mathematical results all tend to provide confirmation of this picture. In the case of black strings it was suggested in a 1988 paper of Gregory and Laflamme that they are also stable on the basis of a linearized analysis. This conclusion was reversed in a 1993 paper of the same authors and the ‘Gregory-Laflamme instability’ was born. It occurs when L is too large in comparison with the mass m. The defect in the analysis of the earlier paper seems to be related to the boundary conditions imposed on the perturbation and their interaction with the mode decomposition which was also used. Once this instability had been discovered it was natural to ask what the endpoint of the evolution of a solution is which originally starts as a growing small perturbation of the black string. The subject of the talk of Pretorius was this question in dimension five. One possibility would be a modification of the standard black string with a characterstic radius depending on the extra direction. This kind of object has been called a non-uniform black string (NUBS) and there is some evidence that objects of this kind exist. Another possibility is that the cylindrical configuration could break up into a row of black holes. Calculations from a few years ago showed this kind of thing but were not definitive since the calculation broke down too soon. Today Pretorius showed impressive numerical results where the calculation runs much longer and reveals surprising phenomena. The diagnostic quantity which is plotted is the radius of the apparent horizon as a function of the extra spatial coordinate and time. To start with some spherical lumps appear in the string and these become more and more pronounced with the part of the string between the lumps becoming thinner and thinner. Being thinner corresponds to a smaller mass and increased susceptibility to the Gregory-Laflamme instability. As time passes new lumps appear and grow to a certain size. The number of these grows larger and larger and it may be that this process takes place infinitely often within a finite time interval. The pictures were beautiful but rather frightening in their complexity. It was mentioned that relations have been made between this process and the Rayleigh-Plateau instability which occurs in the break-up of a stream of water into individual drops.


Cytokine dynamics

June 16, 2010

In a previous post I discussed a paper of Lev Bar-Or on the dynamics of the interactions between T cells and macrophages. More precisely, the unknowns in the dynamical system used in that work are concentrations of cytokines produced by those cells. The object of study is a four-dimensional dynamical system depending on eighteen parameters. In the paper only a very few aspects of this system are studied. Two phase portraits are given which are presumably the result of numerical simulations. Some graphs of time evolutions are also shown. There are no theorems. I have now written a paper where I tried to prove as much as possible about the solutions of this system. My motivation for doing so is twofold. On the one hand I am interested in immunology and the contributions which mathematics can make to that field. On the other hand I am interested in exploring the role of rigorous proofs in applications of mathematics. To what extent can proofs be obtained and what insights can they, and the process of obtaining them, give into the original problem?

All the available results indicate that the dynamics of solutions of this system is simple, with all solutions converging to stationary solutions at late times. I was able to prove that for a large open subset of the parameter space there is a unique stationary solution which acts as an attractor for all solutions. Apart from this results were only obtained under strong restrictions on the parameters. There are cases where there are two stable stationary solutions. These correspond biologically to Th1 and Th2 dominance. No evidence was found for cases with a larger number of stable steady states. Numerical experiments also gave no indications of dynamical behaviour more complicated than what has just been described for any values of the parameters.

In studying this system I ran up against barriers which I was not able to overcome. It remains to be seen to what extent these barriers are in some sense intrinsic to the mathematical problem and to what extent they are a consequence of my inexperience with this type of dynamical system. This is the first time I have written a paper in the area of mathematical biology and so it is a personal landmark for me.

Everything I found is consistent with the idea that the inclusion of such a large number of parameters in the model is superfluous. All types of qualitative behaviour shown by the general system already seem to occur for simpler models with a greatly reduced set of parameters. Moreover switching off some of the major qualitative effects included in the model does not seem to make a big difference. This includes switching off antigen presentation or even leaving out the macrophages completely.