Symmetry. The dictionary defines it as “due or balanced proportions,” and “beauty arising from such harmony.” H. Weyl says it better: “Through symmetry, man tried to perceive and create order, beauty, and perfection.” In both cases, symmetry is seen as equivalent to beauty.
Symmetry has always fascinated humans. Why? It is around us in the physical and biological world. Greeks were obsessed with symmetry and tried to construct an understanding of the heavens in terms of the most symmetrical objects: spheres, and circles. Even Kepler’s formulation of the heliocentric planetary systems was based on symmetry. Humans must have wondered and been enchanted by the many obvious symmetries around them. Trees, plants, animals, and of course humans themselves have obvious symmetries, e.g., left–right. They could not help but ask why these symmetries are present in their attempt to understand nature all around them.
It was not until the 19th century that mathematicians (Galois in the 1820s) formulated and developed group theory. The theory of continuous groups underlies much of our understanding of symmetries such as space groups and crystal structure.
And not until this century was it realized that symmetries are related to conservation laws which are so important in 20th century physics (quantum mechanics). Indeed, symmetries now permeate physics and perhaps other disciplines as well. Moreover, in the past half-century, we have learned that symmetries are not passive, that they do not only restrict the underlying dynamics but, in fact, that they also determine it—i.e., the interaction. It is fascinating to consider this development which ultimately relates physics to geometry. Many of my colleagues believe that the theory of combining all the interactions of nature is likely to be based on a “supersymmetry.”
Symmetry is, indeed, beautiful; but in some sense a perfect symmetry (e.g., sphere) is uninteresting. It becomes more interesting if small asymmetries are present—e.g., the earth is not only not a perfect sphere, but it also has dimples (mountains), etc., which can be understood. During the last half-century, there has been considerable emphasis on “broken symmetries.” In physics, it turns out that even a perfectly symmetric dynamics (theory) can lead to spontaneous breaking at the level of basic states. This phenomenon has been shown to allow the generation of masses in a theory where none are present to begin with. Broken symmetries have become crucial in understanding the world around us.
The purpose of this colloquium is to bring together experts on symmetries from various disciplines to teach us the multi-faceted uses of symmetries. The hope is that the cross-fertilization will lead to new insights and that uses of symmetries in one field may lead to applications in another one.
My hope is that all participants will not only find the Colloquium interesting and stimulating, but also will glean new insights from it.