Polya's conjecture for the Dirichlet Laplacian states that its kth eigenvalue is larger than or equal to the corresponding frst term in the Weyl asymptotics. This conjecture was formulated by Polya in 1954, who then proved it within the class of Euclidean tiling domains in 1961. To this day, there are still very few domains for which the conjecture is known to hold, while the best known results for general Euclidean domains so far are the Berezin-Li-Yau inequalities, obtained by Li and Yau in 1983.
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In this talk we will provide a method for obtaining examples of domains (both Euclidean and non-Euclidean) satisfying Polya's inequality, by relating this to the second term in the Weyl asymptotics. On the other hand, we will also provide examples showing that the nature of this second term and the truth of Polya's conjecture are independent. If time allows, we will show how to improve current known bounds in the case of Euclidean cylinders of the form IxA, where I is a bounded interval and A a bounded n-dimensional Euclidean domain.