selberg trace formula and quantum-classical correspondence in physics

In this note we try to explain the main mathematical ingredients of the selberg trace formula and pose the formula explicitly and survey the main ideas behind this beautiful formula in the compact case. the selberg trace formula is not only confined in the realm of mathematics but it has many applications in theoretical physics. we see that this formula has an interesting interpretation which relates classical mechanical entities like volume, shape, geodesics of the universe to the quantum mechanical entities like eigenvalues (frequencies),eigenfunctions, resonances of the underlying geometry (universe). a little background of complex analysis and hyperbolic geometry is assumed.