Introduction to Trace Formula
The Trace Formula can be understood roughly as an equation relating spectral data to geometric information. It is obtained via expansion of the trace of certain operators that are associated to the Representation Theory of an affine algebraic group, justifying its name. Therefore, the spectral side of the expansion by nature contains data of arithmetic interests. However, the spectral side is generally less accessible. Meanwhile, the geometric side consists of terms that can be written in a more explicit fashion. The computation of the geometric side, which is now referred to as the Orbital Integrals, thus come on the scene. In this talk, we plan to briefly introduce the general derivation of the (vaguely described) Trace Formula, and demonstrate a few concrete examples of it.