Information Services Office   19.10.2011

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Newsletter No. 385 > In Plain View > Laws in Beauty: the Magic of Maths

Laws in Beauty: the Magic of Maths

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Some of you may remember that at the beginning of A Beautiful Mind, a film about the legendary mathematician John Nash, the male protagonist positions a drinking glass against the sun so that the refracted light falls on the necktie of his schoolmate. The budding mathematician muses, 'There could be a mathematical explanation for how bad your tie is.'

Innocent as this quip may seem, Prof. Wei Juncheng of the Department of Mathematics at CUHK would not hesitate to add that natural phenomena can also be explained in mathematical terms and mathematics is the tool for everything ranging from physics, life to finance. The refraction of light in the above example can be explained by the Maxwell Equations. The spots on deer and leopards, and the stripes on clownfish and zebras are all translatable into a set of partial differential equations for the Reaction-Diffusion Systems. Partial differential equations come in handy because a differential operator is needed to express the relationship of a variable with time and space.

Let's look at the animals. Whether we find spots or stripes on their bodies is a result of the distribution and concentration of the molecules of a pigment in the medium of another pigment and the interaction between their molecules. Reaction-diffusion equations can explain the interaction between two media, in different proportions, and how one medium is distributed or aggregates in another medium. They can be used to explain the spots or the stripes found in animals, or even the moles and spots on human skin that may change colour or migrate with time.

When the proportion of two media is more balanced, the mathematical solution of the reaction-diffusion equation would just result in stripes. When the proportion is more extreme, the result would be spots. Thus, both stripes and spots can be described with a pair of reaction-diffusion equations. In other words, the pigments of orange and black on a clownfish are balanced in proportion and result in the stripes we see on it. If the orange pigment dominates, the equation would tell us that the black pigment would aggregate in the form of spots and we would have a clownfish with leopard-like spots. However, simple physical phenomena can be easily explained by relatively simple mathematical equations. With the increasing complexity of other phenomena, more sophisticated mathematical methods are required.

Prof. Wei Juncheng is one of the leading figures in the field of partial differential equations, particularly in the analysis of concentration phenomena in nonlinear elliptic equations and systems. The de Giorgi Conjecture is one of the most famous conjectures in pure mathematics, proposed by the Italian mathematician Ennio de Giorgi in 1978. It concerns the structure of certain nonlinear equations and had puzzled mathematicians around the world. Up to 2006, the conjecture was shown to be true in the second to the eighth dimensions. Thus researchers generally held that the conjecture would apply in any number of dimensions. Through an ingenious mathematical method, Professor Wei and his team were able to find a counter example in the ninth dimension and show that it could not have applied in any dimension higher than the eighth.

Professor Wei's solving of the de Giorgi Conjecture also helped solve the complex reaction-diffusion equations. He found that with the increase of the diffusion parameter in the equations, the spots would become unstable and might split into two or even more to form complex patterns. The seemingly random and haphazard phenomena in nature actually obey rules and follow orderly paths developed with the intelligence of the human mind.

The study of mathematics has made us see through a natural phenomenon. It has also demonstrated its generality of application and structural beauty. The reaction-diffusion equations are not only applicable to animal patterns but also to the physical structure of superconductors, to an understanding of the spread of an epidemic, or may even tell us how cells take in nutrients. Researching into the mathematical structure of differential equations can take one outside mathematics itself into different fields of knowledge. This is what fascinates mathematicians like Prof. Wei Juncheng.

Professor Wei's solving of the de Giorgi Conjecture also helped solve the complex reaction-diffusion equations. He found that with the increase of the diffusion parameter in the equations, the spots would become unstable and might split into two or even more to form complex patterns. The seemingly random and haphazard phenomena in nature actually obey rules and follow orderly paths developed with the intelligence of the human mind.

The study of mathematics has made us see through a natural phenomenon. It has also demonstrated its generality of application and structural beauty. The reaction-diffusion equations are not only applicable to animal patterns but also to the physical structure of superconductors, to an understanding of the spread of an epidemic, or may even tell us how cells take in nutrients. Researching into the mathematical structure of differential equations can take one outside mathematics itself into different fields of knowledge. This is what fascinates mathematicians like Prof. Wei Juncheng.

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