Thanks to you, the speakers and participants, this has been a wonderful Colloquium. I want to thank the speakers for their efforts to make their talks comprehensible to a wide audience of scientists, and all the participants for asking sharp and interesting questions. This made the discussions very lively.
We heard about a wealth and diversity of roles, uses, and applications of symmetries and symmetry-breaking in a large variety of fields: from time-reversible fluid mixing at very low Reynolds numbers in biology to fractals in geophysics and geology. Before the Colloquium, I thought that I might be able to pull the material together, but after listening to the talks, I realized that this was an impossible task. Thus, I will only briefly summarize some of the highlights for me, personally.
Symmetry really became well-defined, mathematically, with Galois in the 1820s. From then on, it developed steadily with Bravais and his classification of lattices and with Lie and his introduction of continuous groups and their finite number of generators. There is a 1:1 correspondence between simply connected Lie groups and highly symmetrical crystals. Noether in the early 1900s showed how symmetries lead to conservation laws.
In biology, we had a lovely demonstration of bacterial motion in low Reynolds number fluids and the uses of cooperation and asymmetry (helices) in this motion.
In evolution, we heard about the complementarity of diversity and complexity. Those genetic changes that replicate more readily are favored. The evolution of bilateral versus radial or helical symmetry are undoubtedly related to motion, but began accidentally.
In astronomy, we learned about deviations from uniformity and the appearance of large-scale structures. Here the first necessity is to map the sky prior to being able to fully understand the distribution of matter. This is an ongoing effort. On the other hand, we heard that gamma ray bursts are distributed uniformly throughout the sky. We also had a guided tour of stellar objects and their symmetries—spherical due to gravity or flattened by gravity combined with rotation, or still more complex symmetries. Strong magnetic fields were shown to cause dipolar symmetries.
In biomacromolecular systems (e.g., insulin), we heard about various symmetrical arrangements: pentamers, hexamers, etc. We learned that the basic reasons for these symmetries are stability, cooperativity, and multivalency.
When it comes to proteins we heard about the huge variety of possibilities; however, nature selects only a few. Why and how? Is it an accident or is it based on symmetry considerations? There seems to be a funnel-like effect in moving towards a unique or semi-unique folding structure. The principle of “minimal frustration” was given as a possible answer; more symmetry may mean less frustration. At the level of viruses we learned of the variety of ways of assembling pentamers into icosohedral structures.
We had a beautiful historical summary of symmetry arguments in chemistry and particularly in molecules.
In physics, we were treated to a wonderful demonstration of quasiperiodic crystals and their relationship to Penrose tilings. We even learned why they form in terms of two or more structural units, matching rules and nonlocality, but were assured that even these features were not necessary; namely, quasicrystals can occur with a single structure by changing or maximizing density, and that local growth rules or interactions are sufficient. We also heard a historical development of symmetry in physics, originally following from the laws of physics and ending by leading to the laws of physics. For instance, permutation symmetry leads to Bose–Einstein or Fermi–Dirac statistics. We were introduced to local gauge (geometrical) symmetries, which dictate the interaction, e.g., the electromagnetic interaction, but do not lead to new conservation laws. All fundamental symmetries are presently believed to be local gauge symmetries. Symmetry breaking is of several varieties and these were introduced.
In animals, bilateral symmetry is obvious and is related to mobility. Is it guaranteed? We learned that small asymmetries appear under environmental stress during development. Although this is reversible, there is also a bias of selection that helps to keep the symmetry. However, there are instances where usage leads to left–right asymmetries as in the crab and our own handedness. But aside from such instances, biases towards asymmetry do not work. The left–right symmetry is amazingly stable.
These aspects are but a small sample of what we heard. There were many other interesting facets of symmetries and asymmetries in different scientific fields that were presented. Time is unfortunately too short for me to summarize them, but we all enjoyed hearing about these multifaceted usages of symmetries.