U.S.-France Cooperative Research: Inverse Problems in Spectral Geometry

这个为期三年的奖项将支持美法在数学领域的合作研究，涉及10名美国大学的研究人员，其中包括5名最近获得博士学位的研究人员，以及法国格勒诺布尔的同等数量的研究人员。美国和法国的小组由来自不同机构的参与者组成，将由达特茅斯学院的卡罗琳·戈登和格勒诺布尔大学傅立叶研究所的皮埃尔·贝拉德领导。他们的研究重点是黎曼几何中的逆谱问题。他们将研究黎曼流形上拉普拉斯算子的特征值谱在多大程度上决定了流形的几何形状。这个问题经常被表述为“一个人能听到鼓的形状吗？”构造同谱流形的一般技术；谱刚性问题；以及黎曼流形的谱与几何之间的关系。这一提议将能力互补、兴趣共同的数学家聚集在一起。该项目的成果将在数学的所有领域，例如在信号处理和遥感方面有重要的应用。

This three-year award will support U.S.-France cooperative research in mathematics and involves ten U.S. university-based investigators, including 5 recent doctorates, and an equal number of investigators in Grenoble, France. The American and French groups, consisting of participants from different institutions, will be led by Carolyn Gordon of Dartmouth College and Pierre Berard of the Institut Fourier, University of Grenoble. The focus of their research is inverse spectral problems in Riemannian geometry. They will study the extent to which eigenvalue spectrum of the Laplace operator on a Riemannian manifold determines the geometry of the manifold. This problem is frequently phrased as `Can one hear the shape of a drum.` General techniques for constructing manifolds with the same spectrum; questions of spectral rigidity; and relationships between the spectrum and the geometry of Riemannian manifolds will be addressed. The proposal brings together mathematicians with complementary abilities and mutual interests. The results of the project will have important applications in all fields of mathematics and for example, in signal processing and remote sensing.