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ќќќќќќџџџџGroup Theory: What is it good for?
Hassan Azad
I joined LUMS very recently and have already noticed how students challenge authority. It is quite invigorating. Some very talented students question the value of teaching abstract subjects like Group Theory and the emphasis, in general, on proofs. The argument is that all the proofs are in the recommended reading material and there is no need to reproduce proofs word by word. But reading proofs on ones own demands great self discipline. In any case, a book is not interactive and students do need an experienced teacher to lead them through an intricate proof.
A good classical proof is a culmination of intense hard work, experimentation and conjecture and contains ideas and techniques which are discoveries of exceptional talents. In the classroom we, the teachers, try to streamline and improve the arguments to facilitate transmission of this knowledge to the next generation. Ideally, we should let students discover proofs by experimentation and conjecture but, due to time constraints, this can be done only with very simple ideas. Certainly we, the teachers, can try to incorporate conceptual thinking in homework assignments and strive to balance routine calculations with playing with ideas.
Group Theory is a technique for studying symmetry and a discipline which brims with highly original ideas. It is one of the most abstract and at the same time the most applied part of Mathematics. To appreciate the applications in any depth and to apply it, you first have to learn the language and its basic techniques. Wherever it is applied, it has led to deep insights and dramatic simplifications in calculations.
I now want to give you some concrete examples and I assume that you are familiar with the basic terminology and ideas of group theory. If not, you can look up the Wikipedia article on Group Theory ( HYPERLINK "http://en.wikipedia.org/wiki/Group_theory" http://en.wikipedia.org/wiki/Group_theory).
What is the genesis of this theory? It is the old problem of solving equations. There were formulae acquired over centuries- for solving equations of degrees two, three and four, yet there was no general formula for solving equations of degree five or more. The final breakthrough came through the efforts of Abel and Galois. The ideas which these young mathematicians introduced to crack these problems leave one stunned with their originality and depth. It is one of the most romantic and enigmatic chapter in the history of Mathematics (see http://www-history.mcs.st-andrews.ac.uk/Biographies/Abel.html and HYPERLINK "http://www-history.mcs.st-andrews.ac.uk/Biographies/Galois.html" http://www-history.mcs.st-andrews.ac.uk/Biographies/Galois.html).
All the basic ideas of group theory- like normality and solvability -appear in embryonic form in the work of Abel and Galois. No amount of experimentation could have suggested these ideas. Then, there are the Sylow theorems, which are pillars of the theory of finite groups. Again, no amount of experimentation could have suggested the ideas necessary to secure these theorems. When Sylow came up with these ideas, there was nothing like computers to facilitate computations, which could have provided experimental evidence for these theorems.
Sophus Lie, a fellow countryman of Sylow, had a more practical bent of mind. He was interested in geometry and differential equations. He wanted to do for differential equations what Abel and Galois had done for algebraic equations. Specifically, he wanted to understand what was behind the various adhoc reduction techniques in differential equations, which facilitate their solutions. This led to his invention of the theory of continuous groups- which is basically a group which lives on a geometric object like a curve, a surface or their higher dimensional analogues- and he used these ideas to investigated symmetries of differential equations. He found that existence of symmetries of differential equations leads to reduction in their order. To date, it is the only known method of finding exact solutions of non-linear differential equations systematically.
Another branch of Mathematics which Group Theory continues to illuminate is Differential Geometry. The founders of this subject- Gauss, Riemann and Poincare- were men with great practical interests. Gauss was involved in projects for drawing accurate maps of Hannover, Riemann was interested in problems of Electricity and Magnetism and Poincare in Mechanics and Dynamical Systems. But they were also interested in internal developments of Mathematics. One of the important problems at that time was to understand why Euclid had taken the fifth postulate as an axiom and whether it could be derived from the remaining axioms. It began to dawn gradually that there were perhaps other geometries where one could have more than one line parallel to a given line and passing through a fixed point which is not on the line. This led to the concepts of metrics and manifolds. In particular, Poincare introduced a geometry which lives on the disc and violates the fifth axiom of Euclid. By studying symmetries of this geometry, one can derive all the basic properties of this geometry with remarkable ease. Indeed, all this can now be assigned as exercises at the senior level.
What was the net result of all these abstractions? The framework provided by these abstract geometries was exactly what was needed by Einstein for his Theory of General Relativity. On the other hand, Hermann Weyl came to group theory because of his desire to understand Relativity better. Subsequently, he made everlasting contributions in the theory of Group Representations, which is a basic tool in Chemistry and Quantum Mechanics.
Actually, this dynamic between abstraction and applications is centuries old. The mathematician Apollonius of Perga (born 262 BC in Perga, modern Antalya in Turkey) studied the mathematics of conic sections. Let us see hear in his own words why he studied them:
They are worthy of acceptance for the sake of demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason.
But if he hadnt initiated the study of conic sections, there would perhaps have been no Kepler, no Newton and no satellite motion. In any case, the properties of conic sections and surfaces of revolutions obtained from them have well-known and everywhere present applications: their focusing properties are used for the transport of signals and energy.
Finally, Emmy Noethers first contributions were of an applied nature. She related group generators to higher conservation laws and her ideas continue to be used very much by contemporary physicists. But all her subsequent contributions were in very abstract parts of mathematics- namely, the theory of Rings and Ideals.
In summary, abstraction and applications have always existed side by side. Some researchers have a natural bent for one over the other; some have a taste for both. It would be futile to insist on the merits of one over the other.
Finally, while discussing applications of Mathematics, it is good to keep in mind the question: What is the use of Music, Poetry and Art, in general?
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