This will be a series of two talks wherein I will give an introductory talk on pseudo-differential operators and their symbols in the first talk and I will present my recent results in the second talk. Pseudo-differential operators are a generalization of differential operators on the space of smooth functions on a manifold. One of the most fundamental constructions on a pseudo-differential operator is its symbol. Under some assumptions on the operator, then there is a one to one correspondence between the operators and the symbols. I generalized Michael Ruzhansky's construction of symbolic calculus on compact Lie groups to symbolic calculus of operators acting on the smooth sections of homogeneous vector bundles. In particular, I proved the formula for the parametrix of elliptic operators.

Symbolic calculus on homogeneous vector bundles (Part I)