PDE/Analysis Seminar

Speaker: Elena Kim (Harvard University) Title: Polynomial bounds for eigenfunctions and eigenvalues on random covers of hyperbolic surfaces Abstract: Let 𝑋 be a compact connected orientable hyperbolic surface and let 𝑋𝑛 be a degree 𝑛 random cover. We show that, with high probability, the 𝐿∞ norm of every Laplace eigenfunction on 𝑋𝑛 with bounded eigenvalue decays polyno- mially in 𝑛. Using similar methods, we also show that, with high probability, the distribution of eigenvalues of the Laplacian on 𝑋𝑛 converges to the spec- tral measure of the hyperbolic plane with polynomially decaying error. Our proof relies on the Selberg pre-trace formula and a variant of the polynomial method This is joint work with Zhongkai Tao.