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## The Two-Point Weyl Law on Manifolds without Conjugate Points

### Blake Keeler

#### McGill University, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak2bbc70cf6bd1f70d3abf8451c863d6e9').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy2bbc70cf6bd1f70d3abf8451c863d6e9 = 'bk&#101;&#101;l&#101;r' + '&#64;'; addy2bbc70cf6bd1f70d3abf8451c863d6e9 = addy2bbc70cf6bd1f70d3abf8451c863d6e9 + 'l&#105;v&#101;' + '&#46;' + '&#117;nc' + '&#46;' + '&#101;d&#117;'; var addy_text2bbc70cf6bd1f70d3abf8451c863d6e9 = 'bk&#101;&#101;l&#101;r' + '&#64;' + 'l&#105;v&#101;' + '&#46;' + '&#117;nc' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak2bbc70cf6bd1f70d3abf8451c863d6e9').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy2bbc70cf6bd1f70d3abf8451c863d6e9 + '\'>'+addy_text2bbc70cf6bd1f70d3abf8451c863d6e9+'<\/a>';

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 compact 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 that 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 $\Pi_\lambda$. We also discuss an application of these results to the study of monochromatic random waves.