Spectral Geometry and Inverse Scattering Theory

Hongyu Liu is a Professor and the Associate Head at the Department of Mathematics, City University of Hong Kong. Before taking up the current position, he worked as a Professor and the Associate Head at the Department of Mathematics, Hong Kong Baptist University (2014--2020). Prior to that, he held faculty positions at University of North Carolina, Charlotte, USA (2011--2014), University of Reading, UK (2010/11), and University of Washington, Seattle, USA (2007--2010). He obtained his PhD in Mathematics from The Chinese University of Hong Kong (2007).  His research focuses on the analysis, computations and applications of inverse problems and imaging, wave propagation, partial differential equations, mathematical materials science, scattering theory and spectral theory. He has also been working on the interplay among inverse scattering techniques, bionic learning and artificial intelligence. He has published over peer-reviewed 150 research papers in leading journals, and in addition he has 16 research preprints under review to leading journals. He coauthored one research monograph by Societe Mathematique de France. 

Introduction. -Geometric structures of Laplacian eiegenfunctions.- Geometric structures of Maxwellian eigenfunctions.-  Inverse obstacle and diffraction grating scattering problems.- Path argument for inverse acoustic and electromagnetic obstacle scattering problems.-  Stability for inverse acoustic obstacle scattering problems. - Stability for inverse electromagnetic obstacle scattering problems.-  Geometric structures of Helmholtz's transmission eigenfunctions with general transmission conditions and applications.- Geometric structures of Maxwell's transmission eigenfunctions and applications.- Geometric structures of Lame's transmission eigenfunctions with general ´ transmission conditions and applications.-  Geometric properties of Helmholtz's transmission eigenfunctions induced by curvatures and applications. - Stable determination of an acoustic medium scatterer by a single far-field pattern.- Stable determination of an elastic medium scatterer by a single far-field measurement and beyond.