Abstract: We will first discuss the relation between curvature and L^q norm estimates of spectral projection operators on compact manifolds. In particular, we will demonstrate that the sign of the curvature of a compact space form can be inferred from the growth rate of L^q-norms of L^2-normalized quasimodes. We will also discuss related recent work on the sharp spectral projection estimates on noncompact asymptotically hyperbolic surfaces with negative curvature, including all convex cocompact hyperbolic surfaces. This is based on joint works with Christopher Sogge, Zhongkai Tao and Zhexing Zhang.

Bio: Xiaoqi Huang is currently an assistant professor at Louisiana State University. He obtained his Ph.D. at Johns Hopkins University in 2021 under the supervision of Christopher Sogge. His research focuses on harmonic analysis and partial differential equations.

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