In 1966, Mark Kac posed the famous question “Can you hear the shape of a drum?” Mathematically, this amounts to recovering the geometry of a Riemannian manifold from knowledge of its Laplace spectrum. In the case of strictly convex and smooth bounded planar domains, the problem is very much open. One technique for studying the inverse spectral problem is via the wave trace, a distribution with singular support contained in the length spectrum. The length spectrum is the collection of lengths of closed geodesics, which for planar domains are just periodic billiard orbits. A dual object to study is the resolvent (of the Laplacian), whose trace asymptotics are related via the Payley-Wiener theorem to singularities of the wave trace. In this talk, we introduce the Balian-Bloch-Zelditch method of constructing a parametrix for the resolvent trace via layer potentials. The result is an oscillatory integral to which one can apply the method of stationary phase. A novel feature is the organization of stationary phase coefficients by using graph theory and Feynman diagrams. The resulting formulas can be used to match Maslov indices of orbits and produce cancellations in the wave trace, which shows that the length spectrum and the Laplace spectrum might be inherently distinct objects, at least insofar as the wave trace is concerned.