Hearing the triangles
数学专题报告
报告题目(Title)：Hearing the triangles
报告人(Speaker)：刘晓东 (中科院应用数学所)
地点(Place)：后主楼 1223
时间(Time)：2023年7月6日(周四), 14:00-15:00
邀请人(Inviter)：李海刚
报告摘要
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 a quadrilateral may not be heard.
