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, late in the last 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 provides us with ample evidence of these symmetries. Snowflakes, molecules, and crystals are good examples of objects with a finite number of symmetries. Their existence is not only of pure mathematical interest, but reflects deeply on the physical characteristics of the objects themselves (how molecules bound for example). Continuous symmetries, which were at the centre of Lie's ideas,  have always played an important role into the insight and development of modern Physics. Here Lie theory arises as signs of the postulated symmetry in the laws of nature; for instance rotations and translations in space for classical mechanics, the Heisenberg group for 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 the belief in this principle is so pervasive, that the laws of some of the current unifying theories (like superstrings) are in fact obtained  this way. Outside the hard sciences we are also surrounded by symmetry as Escher's drawings, Bach's fugues, and Da Vinci's studies clearly show. It is thus not an exageration to say that symmetry (in a Platonic sense), is part of our very nature, and that Lie theory affords us a crucial tool for furthering our understanding of this relationship.