Applications of Galois cohomology to infinite dimensional Lie theory

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My work is in Lie theory, so-named after the Norwegian mathematician Sophus Lie (pronounced Lee) who, in the late 19th century, began the study of this particular field of mathematics. At the heart of Lie theory is the idea of symmetry, both discrete and continuous. Nature provide us with ample evidence of discrete symmetries, such as those exhibited by snow flakes, molecules and crystals. The existence of these symmetries is not only of mathematical interest, but reflect deeply on the physical characteristics of the objects themselves (how strongly can molecules bind together, for example). Continuous symmetries, which were at the heart of Lie's ideas, have played and continue to play a fundamental role in modern physics, providing at times decisive insight into a particular problem or theory. Different parts of Lie theory arise as signs of the postulated symmetry in the laws of nature; for instance rotations and translations in space for classical mechanics, the Heisenberg group in quantum mechanics, and the Lorentz group in the case of special relativity. There is an important common philosophical thread weaving through these examples: The laws of nature tend to be as simple as possible once they are circumscribed by a given group of symmetries. In the case of physics this principle is so pervasive that the laws of some of the current unifying theories (like superstrings) are derived (or conjectured) in this way. Outside the hard sciences we are also surrounded by symmetry, as Escher's drawings, Bach's fugues and Da Vinci's studies illustrate. It is thus not an exaggeration to say that symmetry--in the Platonic sense--is part of our very nature. Lie theory affords us a powerful tool for furthering our understanding of this relationship.