Weyl in 1911 proved the famous Weyl's law, which shows the the asymptotic behavior of eigenvalues only depends on the volume of the domain. Courant, Courant-Hilbert, Hormander, Ivrii, Seeley and many others made contributions to improve the asymptotic estimate of the remainder of Weyl's law. Polya in 1954 conjectured the eigenvalues should have a uniform estimate, where later Lieb, Li-Yau, Berezin and many others made efforts to give uniform estimates for the eigenvalues. The Polya conjecture is still wide open. In this report, we shall discuss some recent works regarding quantitative estimates on the remainder of Weyl's law and Polya's conjecture.
