Applications of torsors in algebra and Lie theory

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Understanding symmetry, and how and why it arises in nature, is important in both Mathematics and Physics. The mathematical objects that measure symmetry are called Groups. Lie groups, named after the Norwegian mathematician Sophus Lie who discovered and first studied these objects late in the 19th century, arise naturally as the underlying symmetry of many theories in Physics. The mid 20th century saw the birth of Algebraic Groups, objects that capture the spirit of Lie groups yet are much more universal. Many of the most striking results in contemporary Mathematics make use of algebraic groups. Their origins go back the fundamental work of Weil, Chevalley, Borel, Serre, Grothendieck, Demazure, who in the 1940s and 50s systematically developed the ideas of Lie and Cartan in the context of algebraic geometry. Over the course of subsequent decades the theory of algebraic groups has been used to give a unified treatment of several key ideas of algebra, including the theories of quadratic and hermitian forms, central simple algebras, algebras with involutions and non-associated algebras. The research project centers on understanding the very nature of algebraic groups themselves, and their applications to several areas of Mathematics, as well as Physics. Given an algebraic group one can associate different geometric objects. Of special interest are torsors and projective homogeneous varieties. Recent results in algebra show that future progress in this area of mathematics depends how deeply we can understand their geometric properties. My project concerns on studying properties of these objects. As an application we plan to apply our results to classification of infinite dimensional Lie algebras which appear in Physics.