報告題目：On nodal and generalized singular structures of Laplacian eigenfunctions and applications to inverse scattering problems
報告摘要：In this talk, we present some novel and intriguing findings on the geometric structures of Laplacian eigenfunctions and their deep relationship to the quantitative behaviors of the eigenfunctions in two dimensions. We introduce a new notion of generalized singular lines of the Laplacian eigenfunctions, and carefully study these singular lines and the nodal lines. The studies reveal that the intersecting angle between two of those lines is closely related to the vanishing order of the eigenfunction at the intersecting point. We establish an accurate and comprehensive quantitative characterization of the relationship. Roughly speaking, the vanishing order is generically infinite if the intersecting angle is irrational, and the vanishing order is finite if the intersecting angle is rational. In fact, in the latter case, the vanishing order is the degree of the rationality. The theoretical findings are original and of significant interest in spectral theory. Moreover, they are applied directly to some physical problems of great importance, including the inverse obstacle scattering problem and the inverse diffraction grating problem. It is shown in a certain polygonal setup that one can recover the support of the unknown scatter as well as the surface impedance parameter by finitely many far-field patterns. Indeed, at most two far-field pat- terns are sufficient for some important applications. Unique identifiability by finitely many far-field patterns remains to be a highly challenging fundamental mathematical problem in the inverse scattering theory.
專家簡介：刁懷安，博士畢業于香港城市大學，東北師范大學數學與統計學院副教授，研究方向數值代數與反散射問題，在Mathematics of Computation, BIT, Numerical Linear Algebra with Applications, Linear Algebra and its Applications等國際知名期刊發表科研論文三十余篇；出版學術專著一本；曾主持國家自然科學基金青年基金項目1項，數學天元基金1項，教育部博士點新教師基金1項；現為吉林省工業與應用數學學會第四屆理事會理事,國際線性代數系會會員；曾多次赴普渡大學、麥克馬斯特大學、漢堡工業大學、日本國立信息研究所、香港科技大學、香港浸會大學等高校進行合作研究與學術訪問。據Web of Science顯示他的單篇論文最高被引用51次。