# 学术报告三十三：郭经纬—The two-term Weyl formulas for planar disks and annuli

One of the most important objects in spectral geometry is the eigenvalue counting function, say, of the Dirichlet Laplacian associated with planar domains.

The simplest examples of domains are squares, disks, annuli, etc. It is well-known that for each of these domains its eigenvalue counting function has an asymptotics containing two main terms and a remainder of size $o(\mu)$. (Such an asymptotics is usually called Weyl's law.) To improve the estimate of the remainder term had been one of the most attractive problems in spectral geometry for decades.

In this talk I will first introduce background and the work by Y. Colin de Verdiere on the two-term Weyl formula for planar disks. Then I will explain how to improve his result by using tools from analysis and analytic number theory and how to extend it from disks to annuli. This is our recent work joint with Wolfgang Mueller, Weiwei Wang and Zuoqin Wang.