In 1968, Hormander introduced the notion of Fourier integral operators and used techniques of microlocal analysis to study the asymptotics of wave kernel $U(t) = exp(-itA)$ associated to a classical elliptic self-adjoint pseudo-differential operator A of order 1 on a compact manifold M for small times, and obtained an improved estimate for the spectral function, when $A = \sqrt{-\triangle}$, the square root of Laplace operator on the compact manifold M, this estimate for the spectral function can be used to improve the error term in the Weyl law describing the asymptotics of counting function of eigenvalues of the Laplacian.

In this talk, I will start by introducing pseudo-differential operators on Manifolds, discuss a few essential properties, then sketch a proof of Hormander's estimate for the spectral function using the parametrix for the wave kernel.

Speaker(s): Punya Satpathy (UM)