Infinite dimensional Lie theory and representation theory

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More than a hundred years ago, the Norwegian mathematician Sophus Lie introduced the basic ideas of Lie theory as a means to study problems in the area of differential equations. From then on, Lie theory has emerged to an integral part of contemporary mathematics which interacts with several other areas, such as differential geometry, algebraic geometry, harmonic analysis, mathematical physics, dynamical systems, and number theory, to name a few. The fundamental objects of Lie theory are continuous transformation groups which are called Lie groups, and certain algebraic structures associated to Lie groups which are called Lie algebras. In many branches of physics and mathematics, Lie groups and Lie algebras can be used to describe and better understand continuous symmetries. Generally speaking, what links a Lie group to symmetries of a mathematical or a physical system is an algebraic structure which is called the representation of the Lie group (or its associated Lie algebra). For example, in modern physics, representations of the Lorentz group have been used to understand various properties of elementary particles. The existing interaction between Lie theory and physics reveals yet another aspect of the subject: that the infinite dimensional Lie groups and Lie algebras are as important as the finite dimensional ones. Motivated by the advent of supersymmetry as a theory in particle physics, in the 1970's mathematicians as well as physicists were drawn to study the structure and representations of graded analogues of Lie groups and Lie algebras which are nowadays called Lie supergroups and Lie superalgebras. The idea that underlies these graded concepts is to allow both commuting and anti-commuting coordinates. The introduction of Lie supergroups and Lie superalgebras has led to a whole new domain of active mathematical research. The goal of this proposal is to study the structure theory and representation theory of Lie algebras/Lie groups and Lie superalgebras/Lie supergroups.