Selberg trace formula I

This talk will begin with an introduction to the length spectrum of a compact hyperbolic surface and a derivation of the Length Trace Formula. We will then use this formula to prove Huber’s Theorem, establishing the equivalence between the length spectrum and the eigenvalue spectrum of the Laplacian. Finally, we will examine the trace of the heat kernel and its relationship to the eigenvalues, specifically focusing on the proof of the pretrace formula.