報告題目： Hearing the triangles
報 告 人：劉曉東 研究員 中國科學院數學與系統科學研究院
報告時間：2023年3月31日 10:00-11:00
報告地點：騰訊會議 ID：854-1722-5585
會議鍊接：https://meeting.tencent.com/dm/FGrbTYZp7mze
校内聯系人：刁懷安 diao @ jlu.edu.cn
報告摘要： Since the landmark paper by Marc Kac in 1966, the question "Can one hear the shape of a drum?" has attracted and inspired many mathematicians. This forms the subject of the mathematical discipline called spectral geometry. We introduce a two-step numerical scheme for reconstructing the shape of a triangle by its Dirichlet spectrum. With the help of the asymptotic behavior of the heat trace, the first step is to determine the area, perimeter, and the sum of the reciprocals of the angles of the triangle. The shape is then reconstructed, in the second step, by solving a nonlinear system of equations on the angles using the Newton iterative method. To our best knowledge, this is the first numerical simulation for the classical inverse spectrum problem in the plane. Numerically, we have used only finitely many eigenvalues to reconstruct the triangles. We give a counter example to show that, even if we have infinitely many eigenvalues, the shape of an quadrilateral may not be heard.
報告人簡介：劉曉東，2004年本科畢業于北京師範大學，2010年在中科院數學院博士畢業後留所工作，先後任助理研究員，副研究員和研究員，期間于2011年到2013年在德國卡魯理工大學做洪堡博士後。劉曉東主要研究方向是偏微分方程反問題，特别是基于波散射的數學理論和數值算法。目前，在SIAM系列，JCP, IP等雜志發表40餘篇文章。基于這些工作，先後入選數學院陳景潤未來之星計劃，中科院青年創新促進會和國家高層次人才計劃。
