Application of Galois cohomology to infinite dimensional Lie theory

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Lie theory, which owes its origins to the famous Norwegian mathematician Sophus Lie at the end of the 19th century, has evolved into a major mathematical subject, with far reaching tentacles in many other mathematical areas. A by-product is two central intrinsically related objects, viz., Lie groups and Lie algebras. Groups (and algebras) are present in all of mathematics. One of their most important applications is to detect/describe symmetry and invariance of objects (the latter being the permanence of a property under certain symmetries). The importance of symmetry to many areas of science is paramount. The quintessential examples of this can be found in Einstein's theory of special relativity. The famous formula e=mc2 follows from a simple algebraic manipulation if one assumes that the basic laws of physics have to be invariant under translations and rotations. The Lie groups in question are examples of affine and orthogonal groups. This line of thought is extremely fertile. For example, some of the current exotic theories in particle physics (e.g., any of the superstring theories) derive formulas by assuming a priori that the theory must respect certain symmetries. The Lie algebras encode much of the information of the groups but in an easier language to manipulate. The representations of the Lie algebras, for examples, correspond to elementary particles in certain physical theories. In the area of Geometry, Lie groups and algebras are typically regarded as “continuously varying” objects. For mathematical reasons, based on applications to number theory, starting in the 1950's, new “algebraic” versions of geometry and Lie theory were developed. This was the birth of algebraic geometry and algebraic groups, which went on to amalgamate with the theory of schemes and reductive group schemes, as developed by A. Grothendieck and his collaborators in the late 60's. Some of the deepest mathematical results established over the last three decades (including A. Wiles's proof of Fermat's last theorem) could not have been possible without Grothendiecks's revolutionary “language” of geometry. My work centers in trying to discover connections between Grothendieck's creations and infinite dimensional Lie theory. The “infinite dimensional” part appears naturally in string theory. It is also important from a strictly mathematical point of view. This has furnished powerful new machinery to the study of Lie theory and that led to fruition a number of fundamental results. Recently discovered new connections to other areas of mathematics indicate that the proposed research approach will continue to produce results of the highest scientific standards.