a brief survey on the selberg trace formula (the compact case)

In this note, we aim to elucidate the fundamental mathematical concepts and mathematical ingredients underlying the selberg trace formula. we explicitly present the formula for compact quotients and provide a brief overview of its interpretation in both mathematics and physics. the selberg trace formula transcends the boundary of mathematics, establishing intriguing connections between classical mechanical entities such as volume, shape, and geodesics on a surface and quantum mechanical entities such as eigenvalues (frequencies), eigenfunctions, and resonances of the underlying geometry. a rudimentary understanding of complex analysis and hyperbolic geometry is assumed.