Lie Theory and Representation Theory
Lie theory is a coherent and uniform approach to studying the symmetries of algebraic and geometric objects. One modern incarnation of this approach is representation theory, i.e. the study of algebraic structures through their actions by linear transformations on vector spaces. Lie theory and representation theory thereby play central roles in many areas at the interface of pure mathematics and theoretical physics, including condensed matter physics, gauge theory, quantum field theory, and symplectic geometry.