Title: The Two-Point Weyl Law on Manifolds without Conjugate Points

Abstract: In this talk, we discuss the asymptotic behavior of the spectral function of the Laplace-Beltrami operator on a compact Riemannian manifold $M$ with no conjugate points. The spectral function, denoted $\Pi_\lambda(x,y),$ is defined as the Schwartz kernel of the orthogonal projection from $L^2(M)$ onto the eigenspaces with eigenvalue at most $\lambda^2$. In the regime where $(x,y)$ is restricted to a sufficiently small neighborhood of the diagonal in $M\times M$, we obtain a uniform logarithmic improvement in the remainder of the asymptotic expansion for $\Pi_\lambda$ and its derivatives of all orders. This generalizes a result of B\'erard which established an on-diagonal estimate for $\Pi_\lambda(x,x)$ without derivatives. Furthermore, when $(x,y)$ avoids a compact neighborhood of the diagonal, we obtain the same logarithmic improvement in the standard upper bound for the derivatives of $\Pi_\lambda$ itself. We also discuss an application of these results to the study of monochromatic random waves.

About the speaker: Blake Keeler is a CRM-ISM Postdoctoral Fellow at McGill University.  He received his Ph.D. in 2021 from the University of North Carolina-Chapel Hill (UNC) under the direction of Yaiza Canzani.  Dr. Keeler’s research interests include microlocal analysis, dispersive PDE, and spectral theory.  While at UNC, he was awarded a GAANN Fellowship, a Senior Teaching Fellowship, and received the J. Burton Linker Award for excellence in undergraduate teaching. Dr. Keeler was also a co-founder of UNC’s Directed Reading Program, co-organized the Triangle Area Graduate Mathematics Conference, and served as Vice-President of the AMS Graduate Student Chapter at UNC.