报告题目：On novel geometric structures of Laplacian eigenfunctions in R^3 and applications to inverse problems
报 告 人：刁怀安 东北师范大学数学与统计学院副教授
This is a continued development of our recent work [Cao et al. arXiv:1902.05798, 2019] on the geometric structures of Laplacian eigenfunctions and their applications to inverse scattering problems. We studied in [Cao et al. arXiv:1902.05798, 2019] the analytic behaviour of the Laplacian eigenfunctions at a point where two nodal or generalised singular lines intersect. The results reveal an important intriguing property that the vanishing order of the eigenfunction at the intersecting point is closely related to the rationality of the intersecting angle. In the current paper, we continue this development in three dimensions and study the analytic behaviours of the Laplacian eigenfunctions at places where nodal or generalised singular planes intersect. Compared with the two-dimensional case, the geometric situation is much more complicated, so is the analysis: the intersection of two planes generates an edge corner, whereas the intersection of more than three planes generates a vertex corner. We provide a systematic and comprehensive characterisation of the relation between the analytic behaviours of an eigenfunction at a corner point and the geometric quantities of that corner for all these geometric cases. Moreover, we apply our spectral results to establish some novel unique identifiability results for the geometric inverse problems of recovering the shape as well as the (possible) surface impedance coefficient by the associated scattering far-field measurements.
刁怀安，博士毕业于香港城市大学，东北师范大学数学与统计学院副教授，研究方向数值代数与反散射问题，在Mathematics of Computation, BIT, Numerical Linear Algebra with Applications, Linear Algebra and its Applications等国际知名期刊发表科研论文三十余篇；出版学术专著一本；曾主持国家自然科学基金青年基金项目1项，数学天元基金1项，教育部博士点新教师基金1项；现为吉林省工业与应用数学学会第四届理事会理事,国际线性代数协会会员；曾多次赴普渡大学、麦克马斯特大学、汉堡工业大学、日本国立信息研究所、香港科技大学、香港浸会大学等高校进行合作研究与学术访问。