by Eberhard Zeidler
The theory of dynamic systems ranges from the movements of the planets and comets all the way to processes in the human brain and how epidemics like aids spread. One of the most important applications of mathematics in all conceivable areas is being able to precisely determine the optimal design of processes. Take the example of the space shuttle. When it landed on the moon, it had to be piloted using the minimum amount of fuel while the Apollo capsule had to return to earth heating up the heat shield as little as possible. It was not possible to test that experimentally, so that it had to be calculated on computers with the methods of optimal control developed by the blind Russian mathematician Lew Pontrjagin (1908-1988) in 1955. All of the fundamental physical processes function so that the action, i.e. the product of energy and time, must be a minimum (or at least critical). In the framework of his search for a universal law of radiation, Max Planck (1858-1947) made the epoch-making discovery in 1900 that there exists a smallest amount of action in nature -- Planck's quantum of action -- and he received the Nobel Prize in Physics in 1918 for this. Later physicists and mathematicians developed general methods to construct theories with discrete action from theories with continuous action. This is a process we call quantization and we encounter mathematical objects here (take vectors in infinite-dimensional Hilbert spaces) that have both the nature of a wave and a particle. The prototype for this are Einstein's light quanta (photons) that he received the Nobel Prize in Physics in 1921 for.
It is only possible to have such an abundance of shapes and forms in nature because there are stable states. There are significant jumps in the time evolution of a system when the system loses its stability when acted upon by an outside force. The bifurcation theory enables mathematics to calculate the limits of stability and the resulting new structures. That is very closely related to phase transitions, an area of intensive research in physics and mathematics. When water cools off, it makes a transition from water to ice, creating bizarre flowery patterns made of ice on your window. But when the universe cooled off after the Big Bang, there were a series of phase transitions and physicists believe that the originally uniform force was broken down into a strong, weak and electromagnetic interaction. The mathematics of symmetry, also called group theory by mathematicians, is very important in developing an understanding of our world. In 1918, the mathematician Emmy Noether (1882-1935) published a celebrated paper, in which she stated that conservation laws follow from symmetries. For example, the theorem of the conservation of energy applies to any physical system subject to laws that remain the same when time is shifted. This holds for the motions of the planets. The standard model of elementary particles maintains that our world consists of 12 basic particles: 6 quarks and 6 leptons (such as the electron and the neutrino). There are also their antiparticles (such as the positron as the antiparticle of the electron). The interactions between these 12 basic particles are mediated with 12 exchange particles: the massless photon (light), three heavy-mass vector bosons (for such things as radioactive decay) and eight massless gluons (for such things as nuclear forces). The Higg's hypothetical particle is responsible for the mass of the vector bosons.
In 1964, the American physicist Murray Gell-Mann (born in 1929) predicted that the proton consists of three quark particles based upon mathematical symmetry deliberations. Gell-Mann received the Nobel Prize in Physics in 1969 for his theory of unitary symmetry. This type of symmetry is difficult to imagine since it is not something we experience in everyday life, but we encounter it in the world of elementary particles. Unitary symmetry uses the imaginary number i in its mathematical deliberations. In contrast to real numbers, it possesses the remarkable property that
The Italian mathematician Raffael Bombielli introduced it formally in 1550 to be able to solve certain equations. We often notice in the history of mathematics that constructions initially introduced for purely innermathematical considerations later develop surprising applications. Take the example of Fermat's theorem of number theory over three hundred and fifty years old that we still use today to very simply and reliably encode information in banks. Gauss said:
Science should be the friend of applications, not its slave.
A number of phenomena in nature and technology involve a violation of symmetries, which is also known as breaking of symmetries. The ice flowers on your window have a significantly lower degree of symmetry than the homogeneous water that is not frozen. We can explain the phenomenon of forbidden spectral lines in the spectra of molecules mathematically with symmetry violations. Experiments show that processes of the weak interaction violate reflection symmetry and the reflected process is prohibited for certain processes such as the beta decay of cobalt. Lee and Yang received the Nobel Prize in Physics in 1957 for the corresponding theory of parity violation in the weak interaction. A special example of this is the fact that amino acids may be either right-handed or left-handed. Amazingly enough, there are only left-handed amino acids in living matter. That might have to do with the fact that right-handed amino acids are destroyed by ultraviolet radiation. We believe that the early universe had a fundamental supersymmetry between elementary particles with half-integral and integral spin. This supersymmetry is no longer observed today, however there are hopes of observing relicts of it at the high-energy acceleration experiments on the CERN beginning in 2006 and providing undoubted proof of the still missing Higgs particle for the standard model. One of the important differences between living and non-living matter is the fact that living processes cannot be reversed in time, meaning they are always irreversible and symmetry under time reflection is violated.
The key method for describing nature with mathematics is setting up mathematical models. An efficient model leaves non-essential details out while concentrating on the essentials. Also, it is important for every model to be aware of the limits of its validity. A case in point is the typical energy scale of the model. A feature of complicated processes like phase transitions is the fact that you have to link several scales with one another. The American physicist Kenneth Wilson received the Nobel Prize in Physics in 1982 for his theory of the renormalization group. Changing scales is also important in the mathematical theory of microstructures of materials and when setting up effective computer algorithms (multigrid methods). Quantum chemistry is familiar with Schrödinger's equation, the mathematical equation for all molecules. However, this equation is not very helpful with large molecules since it is not possible to do so many calculations. Quantum chemists created an extraordinary useful rough model in their density functional method and Walter Kohn received the Nobel Prize in Chemistry in 1999 for this together with John Pople. In the next few years, we will be able to receive gravitational waves from space with the methods of quantum optics (laser technology). We are expecting to gather information on the collision of black holes from this new window into space. However, in order to decode this information, it is necessary to have extraordinarily complicated computer simulations for 4-dimensional shock waves. But, our computers and scientific calculations are at their limits of the performance capability with the 3-dimensional shock waves from supersonic aeroplanes. If we combine computer simulations for molecular dynamics with the knowledge of an experienced chemist, we can eliminate costly experiments in chemistry when manufacturing new medicines. There is a great deal of mathematical knowledge on geometric models behind every image on a computer screen.
Developing a useful model is an art that has to be learned. A mathematician is dependant upon experience gained from experiments and the intuition of the natural scientist and engineer. It is not always easy to maintain this dialogue since different ways of thinking come into collision with one another. And it is therefore an important task for the future to educate young people who are capable of not only thinking mathematically, but also scientifically.
In 1984, the American Mathematical Society published a report on the future of mathematics and Arthur Jaffe from Harvard University wrote at that time:
Mathematical research should be as broad and as original as possible, with very long-range goals. We expect history to repeat itself: we expect that the most profound and useful future applications of mathematics cannot be predicted today, since they will arise from mathematics yet to be discovered.
As with every science, mathematics also has its limits and only religion is in a position to give answers to the profound existential questions of mankind. The quotation of the German poet, Johann Wolfgang von Goethe on the Harnack House of the Max Planck Society in the Dahlem suburb of Berlin sums it up as follows: "The greatest joy of a thinking man is to have explored the explorable and just to admire the unexplorable."
About the Autor
Prof. Dr. Eberhard Zeidler wurde 1940 in Leipzig geboren. Dort studierte er Mathematik und Physik. 1974 wurde er zum ordentlichen Professor für Analysis an die Universität Leipzig berufen. Zusammen mit Prof. Dr. Jürgen Jost und Prof. Dr. Stefan Müller gründete er 1996 das Max-Planck-Institut für Mathematik in den Naturwissenschaften in Leipzig und war von 1996 bis 2003 dessen geschäftsführender Direktor. Er ist Mitglied der Deutschen Akademie der Naturforscher Leopoldina. Für sein Lebenswerk erhielt er den Alfried Krupp Wissenschaftspreis 2006 der Alfried Krupp von Bohlen und Halbach-Stiftung.
Prof. Zeidler starb im November 2016.
Figure 1: Einstein's Cross